Calculus Examples

$f (x) = - x^{4} + 3 x^{2} + 2 x$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $- x^{4} + 3 x^{2} + 2 x$ with respect to $x$ is $\frac{d}{d x} [- x^{4}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2 x]$ .

$\frac{d}{d x} [- x^{4}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2 x]$

Step 1.2

Evaluate $\frac{d}{d x} [- x^{4}]$ .

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Step 1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{4}$ with respect to $x$ is $- \frac{d}{d x} [x^{4}]$ .

$- \frac{d}{d x} [x^{4}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2 x]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$- (4 x^{3}) + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2 x]$

Step 1.2.3

Multiply $4$ by $- 1$ .

$- 4 x^{3} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2 x]$

Step 1.3

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$- 4 x^{3} + 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x]$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- 4 x^{3} + 3 (2 x) + \frac{d}{d x} [2 x]$

Step 1.3.3

Multiply $2$ by $3$ .

$- 4 x^{3} + 6 x + \frac{d}{d x} [2 x]$

Step 1.4

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 1.4.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$- 4 x^{3} + 6 x + 2 \frac{d}{d x} [x]$

Step 1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 4 x^{3} + 6 x + 2 \cdot 1$

Step 1.4.3

Multiply $2$ by $1$ .

$- 4 x^{3} + 6 x + 2$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- 4 x^{3} + 6 x + 2$ with respect to $x$ is $\frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [2]$ .

$f'' (x) = \frac{d}{d x} (- 4 x^{3}) + \frac{d}{d x} (6 x) + \frac{d}{d x} (2)$

Step 2.2

Evaluate $\frac{d}{d x} [- 4 x^{3}]$ .

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Step 2.2.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{3}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = - 4 \frac{d}{d x} x^{3} + \frac{d}{d x} (6 x) + \frac{d}{d x} (2)$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = - 4 (3 x^{2}) + \frac{d}{d x} (6 x) + \frac{d}{d x} (2)$

Step 2.2.3

Multiply $3$ by $- 4$ .

$f'' (x) = - 12 x^{2} + \frac{d}{d x} (6 x) + \frac{d}{d x} (2)$

Step 2.3

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 2.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$f'' (x) = - 12 x^{2} + 6 \frac{d}{d x} (x) + \frac{d}{d x} (2)$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = - 12 x^{2} + 6 \cdot 1 + \frac{d}{d x} (2)$

Step 2.3.3

Multiply $6$ by $1$ .

$f'' (x) = - 12 x^{2} + 6 + \frac{d}{d x} (2)$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$f'' (x) = - 12 x^{2} + 6 + 0$

Step 2.4.2

Add $- 12 x^{2} + 6$ and $0$ .

$f'' (x) = - 12 x^{2} + 6$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- 4 x^{3} + 6 x + 2 = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

By the Sum Rule, the derivative of $- x^{4} + 3 x^{2} + 2 x$ with respect to $x$ is $\frac{d}{d x} [- x^{4}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2 x]$ .

$\frac{d}{d x} [- x^{4}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2 x]$

Step 4.1.2

Evaluate $\frac{d}{d x} [- x^{4}]$ .

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Step 4.1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{4}$ with respect to $x$ is $- \frac{d}{d x} [x^{4}]$ .

$- \frac{d}{d x} [x^{4}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2 x]$

Step 4.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$- (4 x^{3}) + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2 x]$

Step 4.1.2.3

Multiply $4$ by $- 1$ .

$- 4 x^{3} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2 x]$

Step 4.1.3

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 4.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$- 4 x^{3} + 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x]$

Step 4.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- 4 x^{3} + 3 (2 x) + \frac{d}{d x} [2 x]$

Step 4.1.3.3

Multiply $2$ by $3$ .

$- 4 x^{3} + 6 x + \frac{d}{d x} [2 x]$

Step 4.1.4

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 4.1.4.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$- 4 x^{3} + 6 x + 2 \frac{d}{d x} [x]$

Step 4.1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 4 x^{3} + 6 x + 2 \cdot 1$

Step 4.1.4.3

Multiply $2$ by $1$ .

$f' (x) = - 4 x^{3} + 6 x + 2$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- 4 x^{3} + 6 x + 2$ .

$- 4 x^{3} + 6 x + 2$

Step 5

Set the first derivative equal to $0$ then solve the equation $- 4 x^{3} + 6 x + 2 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- 4 x^{3} + 6 x + 2 = 0$

Step 5.2

Factor the left side of the equation.

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Step 5.2.1

Factor $- 2$ out of $- 4 x^{3} + 6 x + 2$ .

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Step 5.2.1.1

Factor $- 2$ out of $- 4 x^{3}$ .

$- 2 (2 x^{3}) + 6 x + 2 = 0$

Step 5.2.1.2

Factor $- 2$ out of $6 x$ .

$- 2 (2 x^{3}) - 2 (- 3 x) + 2 = 0$

Step 5.2.1.3

Factor $- 2$ out of $2$ .

$- 2 (2 x^{3}) - 2 (- 3 x) - 2 \cdot - 1 = 0$

Step 5.2.1.4

Factor $- 2$ out of $- 2 (2 x^{3}) - 2 (- 3 x)$ .

$- 2 (2 x^{3} - 3 x) - 2 \cdot - 1 = 0$

Step 5.2.1.5

Factor $- 2$ out of $- 2 (2 x^{3} - 3 x) - 2 (- 1)$ .

$- 2 (2 x^{3} - 3 x - 1) = 0$

Step 5.2.2

Factor.

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Step 5.2.2.1

Factor $2 x^{3} - 3 x - 1$ using the rational roots test.

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Step 5.2.2.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1$

$q = \pm 1, \pm 2$

Step 5.2.2.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.5$

Step 5.2.2.1.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 5.2.2.1.3.1

Substitute $- 1$ into the polynomial.

$2 {(- 1)}^{3} - 3 \cdot - 1 - 1$

Step 5.2.2.1.3.2

Raise $- 1$ to the power of $3$ .

$2 \cdot - 1 - 3 \cdot - 1 - 1$

Step 5.2.2.1.3.3

Multiply $2$ by $- 1$ .

$- 2 - 3 \cdot - 1 - 1$

Step 5.2.2.1.3.4

Multiply $- 3$ by $- 1$ .

$- 2 + 3 - 1$

Step 5.2.2.1.3.5

Add $- 2$ and $3$ .

$1 - 1$

Step 5.2.2.1.3.6

Subtract $1$ from $1$ .

$0$

Step 5.2.2.1.4

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} - 3 x - 1}{x + 1}$

Step 5.2.2.1.5

Divide $2 x^{3} - 3 x - 1$ by $x + 1$ .

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Step 5.2.2.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$2 x^{3}$

$0 x^{2}$

$3 x$

$1$

Step 5.2.2.1.5.2

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $x$ .

					$2 x^{2}$
	$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$

Step 5.2.2.1.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			+	$2 x^{3}$	+	$2 x^{2}$

Step 5.2.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} + 2 x^{2}$

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$

Step 5.2.2.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$

Step 5.2.2.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$

Step 5.2.2.1.5.7

Divide the highest order term in the dividend $- 2 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	-	$2 x$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$

Step 5.2.2.1.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$2 x$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					-	$2 x^{2}$	-	$2 x$

Step 5.2.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{2} - 2 x$

				$2 x^{2}$	-	$2 x$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$2 x$

Step 5.2.2.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$	-	$2 x$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$2 x$
							-	$x$

Step 5.2.2.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$2 x^{2}$	-	$2 x$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$2 x$
							-	$x$	-	$1$

Step 5.2.2.1.5.12

Divide the highest order term in the dividend $- x$ by the highest order term in divisor $x$ .

				$2 x^{2}$	-	$2 x$	-	$1$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$2 x$
							-	$x$	-	$1$

Step 5.2.2.1.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$2 x$	-	$1$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$2 x$
							-	$x$	-	$1$
							-	$x$	-	$1$

Step 5.2.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- x - 1$

				$2 x^{2}$	-	$2 x$	-	$1$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$2 x$
							-	$x$	-	$1$
							+	$x$	+	$1$

Step 5.2.2.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$	-	$2 x$	-	$1$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$2 x$
							-	$x$	-	$1$
							+	$x$	+	$1$
										$0$

Step 5.2.2.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$2 x^{2} - 2 x - 1$

Step 5.2.2.1.6

Write $2 x^{3} - 3 x - 1$ as a set of factors.

$- 2 ((x + 1) (2 x^{2} - 2 x - 1)) = 0$

Step 5.2.2.2

Remove unnecessary parentheses.

$- 2 (x + 1) (2 x^{2} - 2 x - 1) = 0$

Step 5.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 1 = 0$

$2 x^{2} - 2 x - 1 = 0$

Step 5.4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 5.4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 5.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5.5

Set $2 x^{2} - 2 x - 1$ equal to $0$ and solve for $x$ .

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Step 5.5.1

Set $2 x^{2} - 2 x - 1$ equal to $0$ .

$2 x^{2} - 2 x - 1 = 0$

Step 5.5.2

Solve $2 x^{2} - 2 x - 1 = 0$ for $x$ .

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Step 5.5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5.5.2.2

Substitute the values $a = 2$ , $b = - 2$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (2 \cdot - 1)}}{2 \cdot 2}$

Step 5.5.2.3

Simplify.

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Step 5.5.2.3.1

Simplify the numerator.

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Step 5.5.2.3.1.1

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 5.5.2.3.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 5.5.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot - 1}}{2 \cdot 2}$

Step 5.5.2.3.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 8}}{2 \cdot 2}$

Step 5.5.2.3.1.3

Add $4$ and $8$ .

$x = \frac{2 \pm \sqrt{12}}{2 \cdot 2}$

Step 5.5.2.3.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 5.5.2.3.1.4.1

Factor $4$ out of $12$ .

$x = \frac{2 \pm \sqrt{4 (3)}}{2 \cdot 2}$

Step 5.5.2.3.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 2}$

Step 5.5.2.3.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{3}}{2 \cdot 2}$

Step 5.5.2.3.2

Multiply $2$ by $2$ .

$x = \frac{2 \pm 2 \sqrt{3}}{4}$

Step 5.5.2.3.3

Simplify $\frac{2 \pm 2 \sqrt{3}}{4}$ .

$x = \frac{1 \pm \sqrt{3}}{2}$

Step 5.5.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 5.5.2.4.1

Simplify the numerator.

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Step 5.5.2.4.1.1

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 5.5.2.4.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 5.5.2.4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot - 1}}{2 \cdot 2}$

Step 5.5.2.4.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 8}}{2 \cdot 2}$

Step 5.5.2.4.1.3

Add $4$ and $8$ .

$x = \frac{2 \pm \sqrt{12}}{2 \cdot 2}$

Step 5.5.2.4.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 5.5.2.4.1.4.1

Factor $4$ out of $12$ .

$x = \frac{2 \pm \sqrt{4 (3)}}{2 \cdot 2}$

Step 5.5.2.4.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 2}$

Step 5.5.2.4.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{3}}{2 \cdot 2}$

Step 5.5.2.4.2

Multiply $2$ by $2$ .

$x = \frac{2 \pm 2 \sqrt{3}}{4}$

Step 5.5.2.4.3

Simplify $\frac{2 \pm 2 \sqrt{3}}{4}$ .

$x = \frac{1 \pm \sqrt{3}}{2}$

Step 5.5.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{3}}{2}$

Step 5.5.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 5.5.2.5.1

Simplify the numerator.

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Step 5.5.2.5.1.1

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 5.5.2.5.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 5.5.2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot - 1}}{2 \cdot 2}$

Step 5.5.2.5.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 8}}{2 \cdot 2}$

Step 5.5.2.5.1.3

Add $4$ and $8$ .

$x = \frac{2 \pm \sqrt{12}}{2 \cdot 2}$

Step 5.5.2.5.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 5.5.2.5.1.4.1

Factor $4$ out of $12$ .

$x = \frac{2 \pm \sqrt{4 (3)}}{2 \cdot 2}$

Step 5.5.2.5.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 2}$

Step 5.5.2.5.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{3}}{2 \cdot 2}$

Step 5.5.2.5.2

Multiply $2$ by $2$ .

$x = \frac{2 \pm 2 \sqrt{3}}{4}$

Step 5.5.2.5.3

Simplify $\frac{2 \pm 2 \sqrt{3}}{4}$ .

$x = \frac{1 \pm \sqrt{3}}{2}$

Step 5.5.2.5.4

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{3}}{2}$

Step 5.5.2.6

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{3}}{2}, \frac{1 - \sqrt{3}}{2}$

Step 5.6

The final solution is all the values that make $- 2 (x + 1) (2 x^{2} - 2 x - 1) = 0$ true.

$x = - 1, \frac{1 + \sqrt{3}}{2}, \frac{1 - \sqrt{3}}{2}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = - 1, \frac{1 + \sqrt{3}}{2}, \frac{1 - \sqrt{3}}{2}$

Step 8

Evaluate the second derivative at $x = - 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 12 {(- 1)}^{2} + 6$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

Raise $- 1$ to the power of $2$ .

$- 12 \cdot 1 + 6$

Step 9.1.2

Multiply $- 12$ by $1$ .

$- 12 + 6$

Step 9.2

Add $- 12$ and $6$ .

$- 6$

Step 10

$x = - 1$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - 1$ is a local maximum

Step 11

Find the y-value when $x = - 1$ .

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Step 11.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = - {(- 1)}^{4} + 3 {(- 1)}^{2} + 2 (- 1)$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Multiply $- 1$ by ${(- 1)}^{4}$ by adding the exponents.

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Step 11.2.1.1.1

Multiply $- 1$ by ${(- 1)}^{4}$ .

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Step 11.2.1.1.1.1

Raise $- 1$ to the power of $1$ .

$f (- 1) = (- 1) {(- 1)}^{4} + 3 {(- 1)}^{2} + 2 (- 1)$

Step 11.2.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 1) = {(- 1)}^{1 + 4} + 3 {(- 1)}^{2} + 2 (- 1)$

Step 11.2.1.1.2

Add $1$ and $4$ .

$f (- 1) = {(- 1)}^{5} + 3 {(- 1)}^{2} + 2 (- 1)$

Step 11.2.1.2

Raise $- 1$ to the power of $5$ .

$f (- 1) = - 1 + 3 {(- 1)}^{2} + 2 (- 1)$

Step 11.2.1.3

Raise $- 1$ to the power of $2$ .

$f (- 1) = - 1 + 3 \cdot 1 + 2 (- 1)$

Step 11.2.1.4

Multiply $3$ by $1$ .

$f (- 1) = - 1 + 3 + 2 (- 1)$

Step 11.2.1.5

Multiply $2$ by $- 1$ .

$f (- 1) = - 1 + 3 - 2$

Step 11.2.2

Simplify by adding and subtracting.

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Step 11.2.2.1

Add $- 1$ and $3$ .

$f (- 1) = 2 - 2$

Step 11.2.2.2

Subtract $2$ from $2$ .

$f (- 1) = 0$

Step 11.2.3

The final answer is $0$ .

$y = 0$

Step 12

Evaluate the second derivative at $x = \frac{1 + \sqrt{3}}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 12 {(\frac{1 + \sqrt{3}}{2})}^{2} + 6$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

Apply the product rule to $\frac{1 + \sqrt{3}}{2}$ .

$- 12 \frac{{(1 + \sqrt{3})}^{2}}{2^{2}} + 6$

Step 13.1.2

Raise $2$ to the power of $2$ .

$- 12 \frac{{(1 + \sqrt{3})}^{2}}{4} + 6$

Step 13.1.3

Cancel the common factor of $4$ .

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Step 13.1.3.1

Factor $4$ out of $- 12$ .

$4 (- 3) \frac{{(1 + \sqrt{3})}^{2}}{4} + 6$

Step 13.1.3.2

Cancel the common factor.

$4 \cdot - 3 \frac{{(1 + \sqrt{3})}^{2}}{4} + 6$

Step 13.1.3.3

Rewrite the expression.

$- 3 {(1 + \sqrt{3})}^{2} + 6$

Step 13.1.4

Rewrite ${(1 + \sqrt{3})}^{2}$ as $(1 + \sqrt{3}) (1 + \sqrt{3})$ .

$- 3 ((1 + \sqrt{3}) (1 + \sqrt{3})) + 6$

Step 13.1.5

Expand $(1 + \sqrt{3}) (1 + \sqrt{3})$ using the FOIL Method.

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Step 13.1.5.1

Apply the distributive property.

$- 3 (1 (1 + \sqrt{3}) + \sqrt{3} (1 + \sqrt{3})) + 6$

Step 13.1.5.2

Apply the distributive property.

$- 3 (1 \cdot 1 + 1 \sqrt{3} + \sqrt{3} (1 + \sqrt{3})) + 6$

Step 13.1.5.3

Apply the distributive property.

$- 3 (1 \cdot 1 + 1 \sqrt{3} + \sqrt{3} \cdot 1 + \sqrt{3} \sqrt{3}) + 6$

Step 13.1.6

Simplify and combine like terms.

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Step 13.1.6.1

Simplify each term.

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Step 13.1.6.1.1

Multiply $1$ by $1$ .

$- 3 (1 + 1 \sqrt{3} + \sqrt{3} \cdot 1 + \sqrt{3} \sqrt{3}) + 6$

Step 13.1.6.1.2

Multiply $\sqrt{3}$ by $1$ .

$- 3 (1 + \sqrt{3} + \sqrt{3} \cdot 1 + \sqrt{3} \sqrt{3}) + 6$

Step 13.1.6.1.3

Multiply $\sqrt{3}$ by $1$ .

$- 3 (1 + \sqrt{3} + \sqrt{3} + \sqrt{3} \sqrt{3}) + 6$

Step 13.1.6.1.4

Combine using the product rule for radicals.

$- 3 (1 + \sqrt{3} + \sqrt{3} + \sqrt{3 \cdot 3}) + 6$

Step 13.1.6.1.5

Multiply $3$ by $3$ .

$- 3 (1 + \sqrt{3} + \sqrt{3} + \sqrt{9}) + 6$

Step 13.1.6.1.6

Rewrite $9$ as $3^{2}$ .

$- 3 (1 + \sqrt{3} + \sqrt{3} + \sqrt{3^{2}}) + 6$

Step 13.1.6.1.7

Pull terms out from under the radical, assuming positive real numbers.

$- 3 (1 + \sqrt{3} + \sqrt{3} + 3) + 6$

Step 13.1.6.2

Add $1$ and $3$ .

$- 3 (4 + \sqrt{3} + \sqrt{3}) + 6$

Step 13.1.6.3

Add $\sqrt{3}$ and $\sqrt{3}$ .

$- 3 (4 + 2 \sqrt{3}) + 6$

Step 13.1.7

Apply the distributive property.

$- 3 \cdot 4 - 3 (2 \sqrt{3}) + 6$

Step 13.1.8

Multiply $- 3$ by $4$ .

$- 12 - 3 (2 \sqrt{3}) + 6$

Step 13.1.9

Multiply $2$ by $- 3$ .

$- 12 - 6 \sqrt{3} + 6$

Step 13.2

Add $- 12$ and $6$ .

$- 6 - 6 \sqrt{3}$

Step 14

$x = \frac{1 + \sqrt{3}}{2}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{1 + \sqrt{3}}{2}$ is a local maximum

Step 15

Find the y-value when $x = \frac{1 + \sqrt{3}}{2}$ .

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Step 15.1

Replace the variable $x$ with $\frac{1 + \sqrt{3}}{2}$ in the expression.

$f (\frac{1 + \sqrt{3}}{2}) = - {(\frac{1 + \sqrt{3}}{2})}^{4} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

Apply the product rule to $\frac{1 + \sqrt{3}}{2}$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{{(1 + \sqrt{3})}^{4}}{2^{4}} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.2

Raise $2$ to the power of $4$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{{(1 + \sqrt{3})}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.3

Use the Binomial Theorem.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1^{4} + 4 \cdot (1^{3} \sqrt{3}) + 6 \cdot (1^{2} {\sqrt{3}}^{2}) + 4 \cdot (1 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4

Simplify each term.

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Step 15.2.1.4.1

One to any power is one.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \cdot (1^{3} \sqrt{3}) + 6 \cdot (1^{2} {\sqrt{3}}^{2}) + 4 \cdot (1 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.2

One to any power is one.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \cdot (1 \sqrt{3}) + 6 \cdot (1^{2} {\sqrt{3}}^{2}) + 4 \cdot (1 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.3

Multiply $4$ by $1$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 6 \cdot (1^{2} {\sqrt{3}}^{2}) + 4 \cdot (1 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.4

One to any power is one.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 6 \cdot (1 {\sqrt{3}}^{2}) + 4 \cdot (1 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.5

Multiply $6$ by $1$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 6 {\sqrt{3}}^{2} + 4 \cdot (1 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 15.2.1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 6 {(3^{\frac{1}{2}})}^{2} + 4 \cdot (1 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 6 \cdot 3^{\frac{1}{2} \cdot 2} + 4 \cdot (1 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 6 \cdot 3^{\frac{2}{2}} + 4 \cdot (1 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.6.4

Cancel the common factor of $2$ .

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Step 15.2.1.4.6.4.1

Cancel the common factor.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 6 \cdot 3^{\frac{2}{2}} + 4 \cdot (1 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.6.4.2

Rewrite the expression.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 6 \cdot 3 + 4 \cdot (1 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.6.5

Evaluate the exponent.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 6 \cdot 3 + 4 \cdot (1 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.7

Multiply $6$ by $3$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 18 + 4 \cdot (1 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.8

Multiply $4$ by $1$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 18 + 4 {\sqrt{3}}^{3} + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.9

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 18 + 4 \sqrt{3^{3}} + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.10

Raise $3$ to the power of $3$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 18 + 4 \sqrt{27} + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.11

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 15.2.1.4.11.1

Factor $9$ out of $27$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 18 + 4 \sqrt{9 (3)} + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.11.2

Rewrite $9$ as $3^{2}$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 18 + 4 \sqrt{3^{2} \cdot 3} + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.12

Pull terms out from under the radical.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 18 + 4 (3 \sqrt{3}) + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.13

Multiply $3$ by $4$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.14

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 15.2.1.4.14.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + {(3^{\frac{1}{2}})}^{4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.14.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + 3^{\frac{1}{2} \cdot 4}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.14.3

Combine $\frac{1}{2}$ and $4$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + 3^{\frac{4}{2}}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.14.4

Cancel the common factor of $4$ and $2$ .

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Step 15.2.1.4.14.4.1

Factor $2$ out of $4$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + 3^{\frac{2 \cdot 2}{2}}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.14.4.2

Cancel the common factors.

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Step 15.2.1.4.14.4.2.1

Factor $2$ out of $2$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + 3^{\frac{2 \cdot 2}{2 (1)}}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.14.4.2.2

Cancel the common factor.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + 3^{\frac{2 \cdot 2}{2 \cdot 1}}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.14.4.2.3

Rewrite the expression.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + 3^{\frac{2}{1}}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.14.4.2.4

Divide $2$ by $1$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + 3^{2}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.4.15

Raise $3$ to the power of $2$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{1 + 4 \sqrt{3} + 18 + 12 \sqrt{3} + 9}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.5

Add $1$ and $18$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{19 + 4 \sqrt{3} + 12 \sqrt{3} + 9}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.6

Add $19$ and $9$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{28 + 4 \sqrt{3} + 12 \sqrt{3}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.7

Add $4 \sqrt{3}$ and $12 \sqrt{3}$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{28 + 16 \sqrt{3}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.8

Cancel the common factor of $28 + 16 \sqrt{3}$ and $16$ .

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Step 15.2.1.8.1

Factor $4$ out of $28$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{4 (7) + 16 \sqrt{3}}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.8.2

Factor $4$ out of $16 \sqrt{3}$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{4 (7) + 4 (4 \sqrt{3})}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.8.3

Factor $4$ out of $4 (7) + 4 (4 \sqrt{3})$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{4 (7 + 4 \sqrt{3})}{16} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.8.4

Cancel the common factors.

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Step 15.2.1.8.4.1

Factor $4$ out of $16$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{4 (7 + 4 \sqrt{3})}{4 \cdot 4} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.8.4.2

Cancel the common factor.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{4 (7 + 4 \sqrt{3})}{4 \cdot 4} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.8.4.3

Rewrite the expression.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 {(\frac{1 + \sqrt{3}}{2})}^{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.9

Apply the product rule to $\frac{1 + \sqrt{3}}{2}$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{{(1 + \sqrt{3})}^{2}}{2^{2}}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.10

Raise $2$ to the power of $2$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{{(1 + \sqrt{3})}^{2}}{4}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.11

Rewrite ${(1 + \sqrt{3})}^{2}$ as $(1 + \sqrt{3}) (1 + \sqrt{3})$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{(1 + \sqrt{3}) (1 + \sqrt{3})}{4}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.12

Expand $(1 + \sqrt{3}) (1 + \sqrt{3})$ using the FOIL Method.

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Step 15.2.1.12.1

Apply the distributive property.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{1 (1 + \sqrt{3}) + \sqrt{3} (1 + \sqrt{3})}{4}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.12.2

Apply the distributive property.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{1 \cdot 1 + 1 \sqrt{3} + \sqrt{3} (1 + \sqrt{3})}{4}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.12.3

Apply the distributive property.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{1 \cdot 1 + 1 \sqrt{3} + \sqrt{3} \cdot 1 + \sqrt{3} \sqrt{3}}{4}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.13

Simplify and combine like terms.

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Step 15.2.1.13.1

Simplify each term.

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Step 15.2.1.13.1.1

Multiply $1$ by $1$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{1 + 1 \sqrt{3} + \sqrt{3} \cdot 1 + \sqrt{3} \sqrt{3}}{4}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.13.1.2

Multiply $\sqrt{3}$ by $1$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{1 + \sqrt{3} + \sqrt{3} \cdot 1 + \sqrt{3} \sqrt{3}}{4}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.13.1.3

Multiply $\sqrt{3}$ by $1$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{1 + \sqrt{3} + \sqrt{3} + \sqrt{3} \sqrt{3}}{4}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.13.1.4

Combine using the product rule for radicals.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{1 + \sqrt{3} + \sqrt{3} + \sqrt{3 \cdot 3}}{4}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.13.1.5

Multiply $3$ by $3$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{1 + \sqrt{3} + \sqrt{3} + \sqrt{9}}{4}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.13.1.6

Rewrite $9$ as $3^{2}$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{1 + \sqrt{3} + \sqrt{3} + \sqrt{3^{2}}}{4}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.13.1.7

Pull terms out from under the radical, assuming positive real numbers.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{1 + \sqrt{3} + \sqrt{3} + 3}{4}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.13.2

Add $1$ and $3$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{4 + \sqrt{3} + \sqrt{3}}{4}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.13.3

Add $\sqrt{3}$ and $\sqrt{3}$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{4 + 2 \sqrt{3}}{4}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.14

Cancel the common factor of $4 + 2 \sqrt{3}$ and $4$ .

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Step 15.2.1.14.1

Factor $2$ out of $4$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{2 \cdot 2 + 2 \sqrt{3}}{4}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.14.2

Factor $2$ out of $2 \sqrt{3}$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{2 \cdot 2 + 2 (\sqrt{3})}{4}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.14.3

Factor $2$ out of $2 \cdot 2 + 2 (\sqrt{3})$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{2 \cdot (2 + \sqrt{3})}{4}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.14.4

Cancel the common factors.

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Step 15.2.1.14.4.1

Factor $2$ out of $4$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{2 \cdot (2 + \sqrt{3})}{2 (2)}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.14.4.2

Cancel the common factor.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{2 \cdot (2 + \sqrt{3})}{2 \cdot 2}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.14.4.3

Rewrite the expression.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + 3 (\frac{2 + \sqrt{3}}{2}) + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.15

Combine $3$ and $\frac{2 + \sqrt{3}}{2}$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + \frac{3 (2 + \sqrt{3})}{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.16

Cancel the common factor of $2$ .

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Step 15.2.1.16.1

Cancel the common factor.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + \frac{3 (2 + \sqrt{3})}{2} + 2 (\frac{1 + \sqrt{3}}{2})$

Step 15.2.1.16.2

Rewrite the expression.

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + \frac{3 (2 + \sqrt{3})}{2} + 1 + \sqrt{3}$

Step 15.2.2

To write $\frac{3 (2 + \sqrt{3})}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + \frac{3 (2 + \sqrt{3})}{2} \cdot \frac{2}{2} + 1 + \sqrt{3}$

Step 15.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 15.2.3.1

Multiply $\frac{3 (2 + \sqrt{3})}{2}$ by $\frac{2}{2}$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + \frac{3 (2 + \sqrt{3}) \cdot 2}{2 \cdot 2} + 1 + \sqrt{3}$

Step 15.2.3.2

Multiply $2$ by $2$ .

$f (\frac{1 + \sqrt{3}}{2}) = - \frac{7 + 4 \sqrt{3}}{4} + \frac{3 (2 + \sqrt{3}) \cdot 2}{4} + 1 + \sqrt{3}$

Step 15.2.4

Combine the numerators over the common denominator.

$f (\frac{1 + \sqrt{3}}{2}) = \frac{- (7 + 4 \sqrt{3}) + 3 (2 + \sqrt{3}) \cdot 2}{4} + 1 + \sqrt{3}$

Step 15.2.5

Simplify the numerator.

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Step 15.2.5.1

Apply the distributive property.

$f (\frac{1 + \sqrt{3}}{2}) = \frac{- 1 \cdot 7 - (4 \sqrt{3}) + 3 (2 + \sqrt{3}) \cdot 2}{4} + 1 + \sqrt{3}$

Step 15.2.5.2

Multiply $- 1$ by $7$ .

$f (\frac{1 + \sqrt{3}}{2}) = \frac{- 7 - (4 \sqrt{3}) + 3 (2 + \sqrt{3}) \cdot 2}{4} + 1 + \sqrt{3}$

Step 15.2.5.3

Multiply $4$ by $- 1$ .

$f (\frac{1 + \sqrt{3}}{2}) = \frac{- 7 - 4 \sqrt{3} + 3 (2 + \sqrt{3}) \cdot 2}{4} + 1 + \sqrt{3}$

Step 15.2.5.4

Apply the distributive property.

$f (\frac{1 + \sqrt{3}}{2}) = \frac{- 7 - 4 \sqrt{3} + (3 \cdot 2 + 3 \sqrt{3}) \cdot 2}{4} + 1 + \sqrt{3}$

Step 15.2.5.5

Multiply $3$ by $2$ .

$f (\frac{1 + \sqrt{3}}{2}) = \frac{- 7 - 4 \sqrt{3} + (6 + 3 \sqrt{3}) \cdot 2}{4} + 1 + \sqrt{3}$

Step 15.2.5.6

Apply the distributive property.

$f (\frac{1 + \sqrt{3}}{2}) = \frac{- 7 - 4 \sqrt{3} + 6 \cdot 2 + 3 \sqrt{3} \cdot 2}{4} + 1 + \sqrt{3}$

Step 15.2.5.7

Multiply $6$ by $2$ .

$f (\frac{1 + \sqrt{3}}{2}) = \frac{- 7 - 4 \sqrt{3} + 12 + 3 \sqrt{3} \cdot 2}{4} + 1 + \sqrt{3}$

Step 15.2.5.8

Multiply $2$ by $3$ .

$f (\frac{1 + \sqrt{3}}{2}) = \frac{- 7 - 4 \sqrt{3} + 12 + 6 \sqrt{3}}{4} + 1 + \sqrt{3}$

Step 15.2.5.9

Add $- 7$ and $12$ .

$f (\frac{1 + \sqrt{3}}{2}) = \frac{5 - 4 \sqrt{3} + 6 \sqrt{3}}{4} + 1 + \sqrt{3}$

Step 15.2.5.10

Add $- 4 \sqrt{3}$ and $6 \sqrt{3}$ .

$f (\frac{1 + \sqrt{3}}{2}) = \frac{5 + 2 \sqrt{3}}{4} + 1 + \sqrt{3}$

Step 15.2.6

Simplify the expression.

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Step 15.2.6.1

Write $1$ as a fraction with a common denominator.

$f (\frac{1 + \sqrt{3}}{2}) = \frac{5 + 2 \sqrt{3}}{4} + \frac{4}{4} + \sqrt{3}$

Step 15.2.6.2

Combine the numerators over the common denominator.

$f (\frac{1 + \sqrt{3}}{2}) = \frac{5 + 2 \sqrt{3} + 4}{4} + \sqrt{3}$

Step 15.2.6.3

Add $5$ and $4$ .

$f (\frac{1 + \sqrt{3}}{2}) = \frac{9 + 2 \sqrt{3}}{4} + \sqrt{3}$

Step 15.2.7

To write $\sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (\frac{1 + \sqrt{3}}{2}) = \frac{9 + 2 \sqrt{3}}{4} + \sqrt{3} \cdot \frac{4}{4}$

Step 15.2.8

Combine fractions.

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Step 15.2.8.1

Combine $\sqrt{3}$ and $\frac{4}{4}$ .

$f (\frac{1 + \sqrt{3}}{2}) = \frac{9 + 2 \sqrt{3}}{4} + \frac{\sqrt{3} \cdot 4}{4}$

Step 15.2.8.2

Combine the numerators over the common denominator.

$f (\frac{1 + \sqrt{3}}{2}) = \frac{9 + 2 \sqrt{3} + \sqrt{3} \cdot 4}{4}$

Step 15.2.9

Simplify the numerator.

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Step 15.2.9.1

Move $4$ to the left of $\sqrt{3}$ .

$f (\frac{1 + \sqrt{3}}{2}) = \frac{9 + 2 \sqrt{3} + 4 \cdot \sqrt{3}}{4}$

Step 15.2.9.2

Add $2 \sqrt{3}$ and $4 \sqrt{3}$ .

$f (\frac{1 + \sqrt{3}}{2}) = \frac{9 + 6 \sqrt{3}}{4}$

Step 15.2.10

The final answer is $\frac{9 + 6 \sqrt{3}}{4}$ .

$y = \frac{9 + 6 \sqrt{3}}{4}$

Step 16

Evaluate the second derivative at $x = \frac{1 - \sqrt{3}}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 12 {(\frac{1 - \sqrt{3}}{2})}^{2} + 6$

Step 17

Evaluate the second derivative.

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Step 17.1

Simplify each term.

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Step 17.1.1

Apply the product rule to $\frac{1 - \sqrt{3}}{2}$ .

$- 12 \frac{{(1 - \sqrt{3})}^{2}}{2^{2}} + 6$

Step 17.1.2

Raise $2$ to the power of $2$ .

$- 12 \frac{{(1 - \sqrt{3})}^{2}}{4} + 6$

Step 17.1.3

Cancel the common factor of $4$ .

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Step 17.1.3.1

Factor $4$ out of $- 12$ .

$4 (- 3) \frac{{(1 - \sqrt{3})}^{2}}{4} + 6$

Step 17.1.3.2

Cancel the common factor.

$4 \cdot - 3 \frac{{(1 - \sqrt{3})}^{2}}{4} + 6$

Step 17.1.3.3

Rewrite the expression.

$- 3 {(1 - \sqrt{3})}^{2} + 6$

Step 17.1.4

Rewrite ${(1 - \sqrt{3})}^{2}$ as $(1 - \sqrt{3}) (1 - \sqrt{3})$ .

$- 3 ((1 - \sqrt{3}) (1 - \sqrt{3})) + 6$

Step 17.1.5

Expand $(1 - \sqrt{3}) (1 - \sqrt{3})$ using the FOIL Method.

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Step 17.1.5.1

Apply the distributive property.

$- 3 (1 (1 - \sqrt{3}) - \sqrt{3} (1 - \sqrt{3})) + 6$

Step 17.1.5.2

Apply the distributive property.

$- 3 (1 \cdot 1 + 1 (- \sqrt{3}) - \sqrt{3} (1 - \sqrt{3})) + 6$

Step 17.1.5.3

Apply the distributive property.

$- 3 (1 \cdot 1 + 1 (- \sqrt{3}) - \sqrt{3} \cdot 1 - \sqrt{3} (- \sqrt{3})) + 6$

Step 17.1.6

Simplify and combine like terms.

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Step 17.1.6.1

Simplify each term.

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Step 17.1.6.1.1

Multiply $1$ by $1$ .

$- 3 (1 + 1 (- \sqrt{3}) - \sqrt{3} \cdot 1 - \sqrt{3} (- \sqrt{3})) + 6$

Step 17.1.6.1.2

Multiply $- \sqrt{3}$ by $1$ .

$- 3 (1 - \sqrt{3} - \sqrt{3} \cdot 1 - \sqrt{3} (- \sqrt{3})) + 6$

Step 17.1.6.1.3

Multiply $- 1$ by $1$ .

$- 3 (1 - \sqrt{3} - \sqrt{3} - \sqrt{3} (- \sqrt{3})) + 6$

Step 17.1.6.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 17.1.6.1.4.1

Multiply $- 1$ by $- 1$ .

$- 3 (1 - \sqrt{3} - \sqrt{3} + 1 \sqrt{3} \sqrt{3}) + 6$

Step 17.1.6.1.4.2

Multiply $\sqrt{3}$ by $1$ .

$- 3 (1 - \sqrt{3} - \sqrt{3} + \sqrt{3} \sqrt{3}) + 6$

Step 17.1.6.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$- 3 (1 - \sqrt{3} - \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}) + 6$

Step 17.1.6.1.4.4

Raise $\sqrt{3}$ to the power of $1$ .

$- 3 (1 - \sqrt{3} - \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}) + 6$

Step 17.1.6.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 3 (1 - \sqrt{3} - \sqrt{3} + {\sqrt{3}}^{1 + 1}) + 6$

Step 17.1.6.1.4.6

Add $1$ and $1$ .

$- 3 (1 - \sqrt{3} - \sqrt{3} + {\sqrt{3}}^{2}) + 6$

Step 17.1.6.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 17.1.6.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- 3 (1 - \sqrt{3} - \sqrt{3} + {(3^{\frac{1}{2}})}^{2}) + 6$

Step 17.1.6.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 3 (1 - \sqrt{3} - \sqrt{3} + 3^{\frac{1}{2} \cdot 2}) + 6$

Step 17.1.6.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$- 3 (1 - \sqrt{3} - \sqrt{3} + 3^{\frac{2}{2}}) + 6$

Step 17.1.6.1.5.4

Cancel the common factor of $2$ .

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Step 17.1.6.1.5.4.1

Cancel the common factor.

$- 3 (1 - \sqrt{3} - \sqrt{3} + 3^{\frac{2}{2}}) + 6$

Step 17.1.6.1.5.4.2

Rewrite the expression.

$- 3 (1 - \sqrt{3} - \sqrt{3} + 3^{1}) + 6$

Step 17.1.6.1.5.5

Evaluate the exponent.

$- 3 (1 - \sqrt{3} - \sqrt{3} + 3) + 6$

Step 17.1.6.2

Add $1$ and $3$ .

$- 3 (4 - \sqrt{3} - \sqrt{3}) + 6$

Step 17.1.6.3

Subtract $\sqrt{3}$ from $- \sqrt{3}$ .

$- 3 (4 - 2 \sqrt{3}) + 6$

Step 17.1.7

Apply the distributive property.

$- 3 \cdot 4 - 3 (- 2 \sqrt{3}) + 6$

Step 17.1.8

Multiply $- 3$ by $4$ .

$- 12 - 3 (- 2 \sqrt{3}) + 6$

Step 17.1.9

Multiply $- 2$ by $- 3$ .

$- 12 + 6 \sqrt{3} + 6$

Step 17.2

Add $- 12$ and $6$ .

$- 6 + 6 \sqrt{3}$

Step 18

$x = \frac{1 - \sqrt{3}}{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{1 - \sqrt{3}}{2}$ is a local minimum

Step 19

Find the y-value when $x = \frac{1 - \sqrt{3}}{2}$ .

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Step 19.1

Replace the variable $x$ with $\frac{1 - \sqrt{3}}{2}$ in the expression.

$f (\frac{1 - \sqrt{3}}{2}) = - {(\frac{1 - \sqrt{3}}{2})}^{4} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2

Simplify the result.

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Step 19.2.1

Simplify each term.

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Step 19.2.1.1

Apply the product rule to $\frac{1 - \sqrt{3}}{2}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{{(1 - \sqrt{3})}^{4}}{2^{4}} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.2

Raise $2$ to the power of $4$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{{(1 - \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.3

Use the Binomial Theorem.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1^{4} + 4 \cdot (1^{3} (- \sqrt{3})) + 6 \cdot (1^{2} {(- \sqrt{3})}^{2}) + 4 \cdot (1 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4

Simplify each term.

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Step 19.2.1.4.1

One to any power is one.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 + 4 \cdot (1^{3} (- \sqrt{3})) + 6 \cdot (1^{2} {(- \sqrt{3})}^{2}) + 4 \cdot (1 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.2

One to any power is one.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 + 4 \cdot (1 (- \sqrt{3})) + 6 \cdot (1^{2} {(- \sqrt{3})}^{2}) + 4 \cdot (1 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.3

Multiply $4$ by $1$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 + 4 (- \sqrt{3}) + 6 \cdot (1^{2} {(- \sqrt{3})}^{2}) + 4 \cdot (1 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.4

Multiply $- 1$ by $4$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 6 \cdot (1^{2} {(- \sqrt{3})}^{2}) + 4 \cdot (1 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.5

One to any power is one.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 6 \cdot (1 {(- \sqrt{3})}^{2}) + 4 \cdot (1 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.6

Multiply $6$ by $1$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 6 {(- \sqrt{3})}^{2} + 4 \cdot (1 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.7

Apply the product rule to $- \sqrt{3}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 6 ({(- 1)}^{2} {\sqrt{3}}^{2}) + 4 \cdot (1 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.8

Raise $- 1$ to the power of $2$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 6 (1 {\sqrt{3}}^{2}) + 4 \cdot (1 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.9

Multiply ${\sqrt{3}}^{2}$ by $1$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 6 {\sqrt{3}}^{2} + 4 \cdot (1 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.10

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 19.2.1.4.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 6 {(3^{\frac{1}{2}})}^{2} + 4 \cdot (1 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.10.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 6 \cdot 3^{\frac{1}{2} \cdot 2} + 4 \cdot (1 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.10.3

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 6 \cdot 3^{\frac{2}{2}} + 4 \cdot (1 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.10.4

Cancel the common factor of $2$ .

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Step 19.2.1.4.10.4.1

Cancel the common factor.

Step 19.2.1.4.10.4.2

Rewrite the expression.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 6 \cdot 3 + 4 \cdot (1 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.10.5

Evaluate the exponent.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 6 \cdot 3 + 4 \cdot (1 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.11

Multiply $6$ by $3$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 + 4 \cdot (1 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.12

Multiply $4$ by $1$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 + 4 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.13

Apply the product rule to $- \sqrt{3}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 + 4 ({(- 1)}^{3} {\sqrt{3}}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.14

Raise $- 1$ to the power of $3$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 + 4 (- {\sqrt{3}}^{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.15

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 + 4 (- \sqrt{3^{3}}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.16

Raise $3$ to the power of $3$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 + 4 (- \sqrt{27}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.17

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 19.2.1.4.17.1

Factor $9$ out of $27$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 + 4 (- \sqrt{9 (3)}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.17.2

Rewrite $9$ as $3^{2}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 + 4 (- \sqrt{3^{2} \cdot 3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.18

Pull terms out from under the radical.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 + 4 (- (3 \sqrt{3})) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.19

Multiply $3$ by $- 1$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 + 4 (- 3 \sqrt{3}) + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.20

Multiply $- 3$ by $4$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + {(- \sqrt{3})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.21

Apply the product rule to $- \sqrt{3}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + {(- 1)}^{4} {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.22

Raise $- 1$ to the power of $4$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 1 {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.23

Multiply ${\sqrt{3}}^{4}$ by $1$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + {\sqrt{3}}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.24

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 19.2.1.4.24.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + {(3^{\frac{1}{2}})}^{4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.24.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{1}{2} \cdot 4}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.24.3

Combine $\frac{1}{2}$ and $4$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{4}{2}}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.24.4

Cancel the common factor of $4$ and $2$ .

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Step 19.2.1.4.24.4.1

Factor $2$ out of $4$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{2 \cdot 2}{2}}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.24.4.2

Cancel the common factors.

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Step 19.2.1.4.24.4.2.1

Factor $2$ out of $2$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{2 \cdot 2}{2 (1)}}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.24.4.2.2

Cancel the common factor.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{2 \cdot 2}{2 \cdot 1}}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.24.4.2.3

Rewrite the expression.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{2}{1}}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.24.4.2.4

Divide $2$ by $1$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{2}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.4.25

Raise $3$ to the power of $2$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 9}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.5

Add $1$ and $18$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{19 - 4 \sqrt{3} - 12 \sqrt{3} + 9}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.6

Add $19$ and $9$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{28 - 4 \sqrt{3} - 12 \sqrt{3}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.7

Subtract $12 \sqrt{3}$ from $- 4 \sqrt{3}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{28 - 16 \sqrt{3}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.8

Cancel the common factor of $28 - 16 \sqrt{3}$ and $16$ .

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Step 19.2.1.8.1

Factor $4$ out of $28$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{4 (7) - 16 \sqrt{3}}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.8.2

Factor $4$ out of $- 16 \sqrt{3}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{4 (7) + 4 (- 4 \sqrt{3})}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.8.3

Factor $4$ out of $4 (7) + 4 (- 4 \sqrt{3})$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{4 (7 - 4 \sqrt{3})}{16} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.8.4

Cancel the common factors.

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Step 19.2.1.8.4.1

Factor $4$ out of $16$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{4 (7 - 4 \sqrt{3})}{4 \cdot 4} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.8.4.2

Cancel the common factor.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{4 (7 - 4 \sqrt{3})}{4 \cdot 4} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.8.4.3

Rewrite the expression.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 {(\frac{1 - \sqrt{3}}{2})}^{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.9

Apply the product rule to $\frac{1 - \sqrt{3}}{2}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{{(1 - \sqrt{3})}^{2}}{2^{2}}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.10

Raise $2$ to the power of $2$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{{(1 - \sqrt{3})}^{2}}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.11

Rewrite ${(1 - \sqrt{3})}^{2}$ as $(1 - \sqrt{3}) (1 - \sqrt{3})$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{(1 - \sqrt{3}) (1 - \sqrt{3})}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.12

Expand $(1 - \sqrt{3}) (1 - \sqrt{3})$ using the FOIL Method.

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Step 19.2.1.12.1

Apply the distributive property.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 (1 - \sqrt{3}) - \sqrt{3} (1 - \sqrt{3})}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.12.2

Apply the distributive property.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 \cdot 1 + 1 (- \sqrt{3}) - \sqrt{3} (1 - \sqrt{3})}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.12.3

Apply the distributive property.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 \cdot 1 + 1 (- \sqrt{3}) - \sqrt{3} \cdot 1 - \sqrt{3} (- \sqrt{3})}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.13

Simplify and combine like terms.

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Step 19.2.1.13.1

Simplify each term.

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Step 19.2.1.13.1.1

Multiply $1$ by $1$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 + 1 (- \sqrt{3}) - \sqrt{3} \cdot 1 - \sqrt{3} (- \sqrt{3})}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.13.1.2

Multiply $- \sqrt{3}$ by $1$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 - \sqrt{3} - \sqrt{3} \cdot 1 - \sqrt{3} (- \sqrt{3})}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.13.1.3

Multiply $- 1$ by $1$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 - \sqrt{3} - \sqrt{3} - \sqrt{3} (- \sqrt{3})}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.13.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 19.2.1.13.1.4.1

Multiply $- 1$ by $- 1$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 - \sqrt{3} - \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.13.1.4.2

Multiply $\sqrt{3}$ by $1$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 - \sqrt{3} - \sqrt{3} + \sqrt{3} \sqrt{3}}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.13.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 - \sqrt{3} - \sqrt{3} + \sqrt{3} \sqrt{3}}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.13.1.4.4

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 - \sqrt{3} - \sqrt{3} + \sqrt{3} \sqrt{3}}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.13.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 - \sqrt{3} - \sqrt{3} + {\sqrt{3}}^{1 + 1}}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.13.1.4.6

Add $1$ and $1$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 - \sqrt{3} - \sqrt{3} + {\sqrt{3}}^{2}}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.13.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 19.2.1.13.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 - \sqrt{3} - \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.13.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 - \sqrt{3} - \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.13.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 - \sqrt{3} - \sqrt{3} + 3^{\frac{2}{2}}}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.13.1.5.4

Cancel the common factor of $2$ .

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Step 19.2.1.13.1.5.4.1

Cancel the common factor.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 - \sqrt{3} - \sqrt{3} + 3^{\frac{2}{2}}}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.13.1.5.4.2

Rewrite the expression.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 - \sqrt{3} - \sqrt{3} + 3}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.13.1.5.5

Evaluate the exponent.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{1 - \sqrt{3} - \sqrt{3} + 3}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.13.2

Add $1$ and $3$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{4 - \sqrt{3} - \sqrt{3}}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.13.3

Subtract $\sqrt{3}$ from $- \sqrt{3}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{4 - 2 \sqrt{3}}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.14

Cancel the common factor of $4 - 2 \sqrt{3}$ and $4$ .

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Step 19.2.1.14.1

Factor $2$ out of $4$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{2 (2) - 2 \sqrt{3}}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.14.2

Factor $2$ out of $- 2 \sqrt{3}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{2 (2) + 2 (- \sqrt{3})}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.14.3

Factor $2$ out of $2 (2) + 2 (- \sqrt{3})$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{2 (2 - \sqrt{3})}{4}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.14.4

Cancel the common factors.

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Step 19.2.1.14.4.1

Factor $2$ out of $4$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{2 (2 - \sqrt{3})}{2 \cdot 2}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.14.4.2

Cancel the common factor.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{2 (2 - \sqrt{3})}{2 \cdot 2}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.14.4.3

Rewrite the expression.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + 3 (\frac{2 - \sqrt{3}}{2}) + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.15

Combine $3$ and $\frac{2 - \sqrt{3}}{2}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + \frac{3 (2 - \sqrt{3})}{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.16

Cancel the common factor of $2$ .

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Step 19.2.1.16.1

Cancel the common factor.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + \frac{3 (2 - \sqrt{3})}{2} + 2 (\frac{1 - \sqrt{3}}{2})$

Step 19.2.1.16.2

Rewrite the expression.

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + \frac{3 (2 - \sqrt{3})}{2} + 1 - \sqrt{3}$

Step 19.2.2

To write $\frac{3 (2 - \sqrt{3})}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + \frac{3 (2 - \sqrt{3})}{2} \cdot \frac{2}{2} + 1 - \sqrt{3}$

Step 19.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 19.2.3.1

Multiply $\frac{3 (2 - \sqrt{3})}{2}$ by $\frac{2}{2}$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + \frac{3 (2 - \sqrt{3}) \cdot 2}{2 \cdot 2} + 1 - \sqrt{3}$

Step 19.2.3.2

Multiply $2$ by $2$ .

$f (\frac{1 - \sqrt{3}}{2}) = - \frac{7 - 4 \sqrt{3}}{4} + \frac{3 (2 - \sqrt{3}) \cdot 2}{4} + 1 - \sqrt{3}$

Step 19.2.4

Combine the numerators over the common denominator.

$f (\frac{1 - \sqrt{3}}{2}) = \frac{- (7 - 4 \sqrt{3}) + 3 (2 - \sqrt{3}) \cdot 2}{4} + 1 - \sqrt{3}$

Step 19.2.5

Simplify the numerator.

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Step 19.2.5.1

Apply the distributive property.

$f (\frac{1 - \sqrt{3}}{2}) = \frac{- 1 \cdot 7 - (- 4 \sqrt{3}) + 3 (2 - \sqrt{3}) \cdot 2}{4} + 1 - \sqrt{3}$

Step 19.2.5.2

Multiply $- 1$ by $7$ .

$f (\frac{1 - \sqrt{3}}{2}) = \frac{- 7 - (- 4 \sqrt{3}) + 3 (2 - \sqrt{3}) \cdot 2}{4} + 1 - \sqrt{3}$

Step 19.2.5.3

Multiply $- 4$ by $- 1$ .

$f (\frac{1 - \sqrt{3}}{2}) = \frac{- 7 + 4 \sqrt{3} + 3 (2 - \sqrt{3}) \cdot 2}{4} + 1 - \sqrt{3}$

Step 19.2.5.4

Apply the distributive property.

$f (\frac{1 - \sqrt{3}}{2}) = \frac{- 7 + 4 \sqrt{3} + (3 \cdot 2 + 3 (- \sqrt{3})) \cdot 2}{4} + 1 - \sqrt{3}$

Step 19.2.5.5

Multiply $3$ by $2$ .

$f (\frac{1 - \sqrt{3}}{2}) = \frac{- 7 + 4 \sqrt{3} + (6 + 3 (- \sqrt{3})) \cdot 2}{4} + 1 - \sqrt{3}$

Step 19.2.5.6

Multiply $- 1$ by $3$ .

$f (\frac{1 - \sqrt{3}}{2}) = \frac{- 7 + 4 \sqrt{3} + (6 - 3 \sqrt{3}) \cdot 2}{4} + 1 - \sqrt{3}$

Step 19.2.5.7

Apply the distributive property.

$f (\frac{1 - \sqrt{3}}{2}) = \frac{- 7 + 4 \sqrt{3} + 6 \cdot 2 - 3 \sqrt{3} \cdot 2}{4} + 1 - \sqrt{3}$

Step 19.2.5.8

Multiply $6$ by $2$ .

$f (\frac{1 - \sqrt{3}}{2}) = \frac{- 7 + 4 \sqrt{3} + 12 - 3 \sqrt{3} \cdot 2}{4} + 1 - \sqrt{3}$

Step 19.2.5.9

Multiply $2$ by $- 3$ .

$f (\frac{1 - \sqrt{3}}{2}) = \frac{- 7 + 4 \sqrt{3} + 12 - 6 \sqrt{3}}{4} + 1 - \sqrt{3}$

Step 19.2.5.10

Add $- 7$ and $12$ .

$f (\frac{1 - \sqrt{3}}{2}) = \frac{5 + 4 \sqrt{3} - 6 \sqrt{3}}{4} + 1 - \sqrt{3}$

Step 19.2.5.11

Subtract $6 \sqrt{3}$ from $4 \sqrt{3}$ .

$f (\frac{1 - \sqrt{3}}{2}) = \frac{5 - 2 \sqrt{3}}{4} + 1 - \sqrt{3}$

Step 19.2.6

Simplify the expression.

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Step 19.2.6.1

Write $1$ as a fraction with a common denominator.

$f (\frac{1 - \sqrt{3}}{2}) = \frac{5 - 2 \sqrt{3}}{4} + \frac{4}{4} - \sqrt{3}$

Step 19.2.6.2

Combine the numerators over the common denominator.

$f (\frac{1 - \sqrt{3}}{2}) = \frac{5 - 2 \sqrt{3} + 4}{4} - \sqrt{3}$

Step 19.2.6.3

Add $5$ and $4$ .

$f (\frac{1 - \sqrt{3}}{2}) = \frac{9 - 2 \sqrt{3}}{4} - \sqrt{3}$

Step 19.2.7

To write $- \sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (\frac{1 - \sqrt{3}}{2}) = \frac{9 - 2 \sqrt{3}}{4} - \sqrt{3} \cdot \frac{4}{4}$

Step 19.2.8

Combine fractions.

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Step 19.2.8.1

Combine $- \sqrt{3}$ and $\frac{4}{4}$ .

$f (\frac{1 - \sqrt{3}}{2}) = \frac{9 - 2 \sqrt{3}}{4} + \frac{- \sqrt{3} \cdot 4}{4}$

Step 19.2.8.2

Combine the numerators over the common denominator.

$f (\frac{1 - \sqrt{3}}{2}) = \frac{9 - 2 \sqrt{3} - \sqrt{3} \cdot 4}{4}$

Step 19.2.9

Simplify the numerator.

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Step 19.2.9.1

Multiply $4$ by $- 1$ .

$f (\frac{1 - \sqrt{3}}{2}) = \frac{9 - 2 \sqrt{3} - 4 \sqrt{3}}{4}$

Step 19.2.9.2

Subtract $4 \sqrt{3}$ from $- 2 \sqrt{3}$ .

$f (\frac{1 - \sqrt{3}}{2}) = \frac{9 - 6 \sqrt{3}}{4}$

Step 19.2.10

The final answer is $\frac{9 - 6 \sqrt{3}}{4}$ .

$y = \frac{9 - 6 \sqrt{3}}{4}$

Step 20

These are the local extrema for $f (x) = - x^{4} + 3 x^{2} + 2 x$ .

$(- 1, 0)$ is a local maxima

$(\frac{1 + \sqrt{3}}{2}, \frac{9 + 6 \sqrt{3}}{4})$ is a local maxima

$(\frac{1 - \sqrt{3}}{2}, \frac{9 - 6 \sqrt{3}}{4})$ is a local minima

Step 21