Calculus Examples

$f (x) = 2 x^{3} + x^{2} - 11 x$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $2 x^{3} + x^{2} - 11 x$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 11 x]$ .

$\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 11 x]$

Step 1.2

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

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Step 1.2.1

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

$2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 11 x]$

Step 1.2.2

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

$2 (3 x^{2}) + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 11 x]$

Step 1.2.3

Multiply $3$ by $2$ .

$6 x^{2} + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 11 x]$

Step 1.3

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

$6 x^{2} + 2 x + \frac{d}{d x} [- 11 x]$

Step 1.4

Evaluate $\frac{d}{d x} [- 11 x]$ .

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Step 1.4.1

Since $- 11$ is constant with respect to $x$ , the derivative of $- 11 x$ with respect to $x$ is $- 11 \frac{d}{d x} [x]$ .

$6 x^{2} + 2 x - 11 \frac{d}{d x} [x]$

Step 1.4.2

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

$6 x^{2} + 2 x - 11 \cdot 1$

Step 1.4.3

Multiply $- 11$ by $1$ .

$6 x^{2} + 2 x - 11$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $6 x^{2} + 2 x - 11$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 11]$ .

$f'' (x) = \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (2 x) + \frac{d}{d x} (- 11)$

Step 2.2

Evaluate $\frac{d}{d x} [6 x^{2}]$ .

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Step 2.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 6 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (2 x) + \frac{d}{d x} (- 11)$

Step 2.2.2

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

$f'' (x) = 6 (2 x) + \frac{d}{d x} (2 x) + \frac{d}{d x} (- 11)$

Step 2.2.3

Multiply $2$ by $6$ .

$f'' (x) = 12 x + \frac{d}{d x} (2 x) + \frac{d}{d x} (- 11)$

Step 2.3

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

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Step 2.3.1

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

$f'' (x) = 12 x + 2 \frac{d}{d x} (x) + \frac{d}{d x} (- 11)$

Step 2.3.2

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

$f'' (x) = 12 x + 2 \cdot 1 + \frac{d}{d x} (- 11)$

Step 2.3.3

Multiply $2$ by $1$ .

$f'' (x) = 12 x + 2 + \frac{d}{d x} (- 11)$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $- 11$ is constant with respect to $x$ , the derivative of $- 11$ with respect to $x$ is $0$ .

$f'' (x) = 12 x + 2 + 0$

Step 2.4.2

Add $12 x + 2$ and $0$ .

$f'' (x) = 12 x + 2$

Step 3

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

$6 x^{2} + 2 x - 11 = 0$

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 $2 x^{3} + x^{2} - 11 x$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 11 x]$ .

$f' (x) = \frac{d}{d x} (2 x^{3}) + \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 11 x)$

Step 4.1.2

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

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Step 4.1.2.1

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

$f' (x) = 2 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 11 x)$

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 $n = 3$ .

$f' (x) = 2 (3 x^{2}) + \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 11 x)$

Step 4.1.2.3

Multiply $3$ by $2$ .

$f' (x) = 6 x^{2} + \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 11 x)$

Step 4.1.3

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

$f' (x) = 6 x^{2} + 2 x + \frac{d}{d x} (- 11 x)$

Step 4.1.4

Evaluate $\frac{d}{d x} [- 11 x]$ .

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Step 4.1.4.1

Since $- 11$ is constant with respect to $x$ , the derivative of $- 11 x$ with respect to $x$ is $- 11 \frac{d}{d x} [x]$ .

$f' (x) = 6 x^{2} + 2 x - 11 \frac{d}{d x} x$

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 $n = 1$ .

$f' (x) = 6 x^{2} + 2 x - 11 \cdot 1$

Step 4.1.4.3

Multiply $- 11$ by $1$ .

$f' (x) = 6 x^{2} + 2 x - 11$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $6 x^{2} + 2 x - 11$ .

$6 x^{2} + 2 x - 11$

Step 5

Set the first derivative equal to $0$ then solve the equation $6 x^{2} + 2 x - 11 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$6 x^{2} + 2 x - 11 = 0$

Step 5.2

Use the quadratic formula to find the solutions.

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

Step 5.3

Substitute the values $a = 6$ , $b = 2$ , and $c = - 11$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (6 \cdot - 11)}}{2 \cdot 6}$

Step 5.4

Simplify.

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Step 5.4.1

Simplify the numerator.

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Step 5.4.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 6 \cdot - 11}}{2 \cdot 6}$

Step 5.4.1.2

Multiply $- 4 \cdot 6 \cdot - 11$ .

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Step 5.4.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{- 2 \pm \sqrt{4 - 24 \cdot - 11}}{2 \cdot 6}$

Step 5.4.1.2.2

Multiply $- 24$ by $- 11$ .

$x = \frac{- 2 \pm \sqrt{4 + 264}}{2 \cdot 6}$

Step 5.4.1.3

Add $4$ and $264$ .

$x = \frac{- 2 \pm \sqrt{268}}{2 \cdot 6}$

Step 5.4.1.4

Rewrite $268$ as $2^{2} \cdot 67$ .

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Step 5.4.1.4.1

Factor $4$ out of $268$ .

$x = \frac{- 2 \pm \sqrt{4 (67)}}{2 \cdot 6}$

Step 5.4.1.4.2

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 67}}{2 \cdot 6}$

Step 5.4.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{67}}{2 \cdot 6}$

Step 5.4.2

Multiply $2$ by $6$ .

$x = \frac{- 2 \pm 2 \sqrt{67}}{12}$

Step 5.4.3

Simplify $\frac{- 2 \pm 2 \sqrt{67}}{12}$ .

$x = \frac{- 1 \pm \sqrt{67}}{6}$

Step 5.5

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

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Step 5.5.1

Simplify the numerator.

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Step 5.5.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 6 \cdot - 11}}{2 \cdot 6}$

Step 5.5.1.2

Multiply $- 4 \cdot 6 \cdot - 11$ .

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Step 5.5.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{- 2 \pm \sqrt{4 - 24 \cdot - 11}}{2 \cdot 6}$

Step 5.5.1.2.2

Multiply $- 24$ by $- 11$ .

$x = \frac{- 2 \pm \sqrt{4 + 264}}{2 \cdot 6}$

Step 5.5.1.3

Add $4$ and $264$ .

$x = \frac{- 2 \pm \sqrt{268}}{2 \cdot 6}$

Step 5.5.1.4

Rewrite $268$ as $2^{2} \cdot 67$ .

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Step 5.5.1.4.1

Factor $4$ out of $268$ .

$x = \frac{- 2 \pm \sqrt{4 (67)}}{2 \cdot 6}$

Step 5.5.1.4.2

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 67}}{2 \cdot 6}$

Step 5.5.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{67}}{2 \cdot 6}$

Step 5.5.2

Multiply $2$ by $6$ .

$x = \frac{- 2 \pm 2 \sqrt{67}}{12}$

Step 5.5.3

Simplify $\frac{- 2 \pm 2 \sqrt{67}}{12}$ .

$x = \frac{- 1 \pm \sqrt{67}}{6}$

Step 5.5.4

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{67}}{6}$

Step 5.5.5

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + \sqrt{67}}{6}$

Step 5.5.6

Factor $- 1$ out of $\sqrt{67}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{67})}{6}$

Step 5.5.7

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{67})$ .

$x = \frac{- 1 (1 - \sqrt{67})}{6}$

Step 5.5.8

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{67}}{6}$

Step 5.6

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

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Step 5.6.1

Simplify the numerator.

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Step 5.6.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 6 \cdot - 11}}{2 \cdot 6}$

Step 5.6.1.2

Multiply $- 4 \cdot 6 \cdot - 11$ .

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Step 5.6.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{- 2 \pm \sqrt{4 - 24 \cdot - 11}}{2 \cdot 6}$

Step 5.6.1.2.2

Multiply $- 24$ by $- 11$ .

$x = \frac{- 2 \pm \sqrt{4 + 264}}{2 \cdot 6}$

Step 5.6.1.3

Add $4$ and $264$ .

$x = \frac{- 2 \pm \sqrt{268}}{2 \cdot 6}$

Step 5.6.1.4

Rewrite $268$ as $2^{2} \cdot 67$ .

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Step 5.6.1.4.1

Factor $4$ out of $268$ .

$x = \frac{- 2 \pm \sqrt{4 (67)}}{2 \cdot 6}$

Step 5.6.1.4.2

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 67}}{2 \cdot 6}$

Step 5.6.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{67}}{2 \cdot 6}$

Step 5.6.2

Multiply $2$ by $6$ .

$x = \frac{- 2 \pm 2 \sqrt{67}}{12}$

Step 5.6.3

Simplify $\frac{- 2 \pm 2 \sqrt{67}}{12}$ .

$x = \frac{- 1 \pm \sqrt{67}}{6}$

Step 5.6.4

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{67}}{6}$

Step 5.6.5

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 - \sqrt{67}}{6}$

Step 5.6.6

Factor $- 1$ out of $- \sqrt{67}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{67})}{6}$

Step 5.6.7

Factor $- 1$ out of $- 1 (1) - (\sqrt{67})$ .

$x = \frac{- 1 (1 + \sqrt{67})}{6}$

Step 5.6.8

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{67}}{6}$

Step 5.7

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{67}}{6}, - \frac{1 + \sqrt{67}}{6}$

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.

$x = - \frac{1 - \sqrt{67}}{6}, - \frac{1 + \sqrt{67}}{6}$

Step 8

Evaluate the second derivative at $x = - \frac{1 - \sqrt{67}}{6}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$12 (- \frac{1 - \sqrt{67}}{6}) + 2$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

Cancel the common factor of $6$ .

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Step 9.1.1.1

Move the leading negative in $- \frac{1 - \sqrt{67}}{6}$ into the numerator.

$12 \frac{- (1 - \sqrt{67})}{6} + 2$

Step 9.1.1.2

Factor $6$ out of $12$ .

$6 (2) \frac{- (1 - \sqrt{67})}{6} + 2$

Step 9.1.1.3

Cancel the common factor.

$6 \cdot 2 \frac{- (1 - \sqrt{67})}{6} + 2$

Step 9.1.1.4

Rewrite the expression.

$2 (- (1 - \sqrt{67})) + 2$

Step 9.1.2

Multiply $- 1$ by $2$ .

$- 2 (1 - \sqrt{67}) + 2$

Step 9.1.3

Apply the distributive property.

$- 2 \cdot 1 - 2 (- \sqrt{67}) + 2$

Step 9.1.4

Multiply $- 2$ by $1$ .

$- 2 - 2 (- \sqrt{67}) + 2$

Step 9.1.5

Multiply $- 1$ by $- 2$ .

$- 2 + 2 \sqrt{67} + 2$

Step 9.2

Simplify by adding numbers.

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Step 9.2.1

Add $- 2$ and $2$ .

$0 + 2 \sqrt{67}$

Step 9.2.2

Add $0$ and $2 \sqrt{67}$ .

$2 \sqrt{67}$

Step 10

$x = - \frac{1 - \sqrt{67}}{6}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{1 - \sqrt{67}}{6}$ is a local minimum

Step 11

Find the y-value when $x = - \frac{1 - \sqrt{67}}{6}$ .

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Step 11.1

Replace the variable $x$ with $- \frac{1 - \sqrt{67}}{6}$ in the expression.

$f (- \frac{1 - \sqrt{67}}{6}) = 2 {(- \frac{1 - \sqrt{67}}{6})}^{3} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 11.2.1.1.1

Apply the product rule to $- \frac{1 - \sqrt{67}}{6}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = 2 ({(- 1)}^{3} {(\frac{1 - \sqrt{67}}{6})}^{3}) + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.1.2

Apply the product rule to $\frac{1 - \sqrt{67}}{6}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = 2 ({(- 1)}^{3} (\frac{{(1 - \sqrt{67})}^{3}}{6^{3}})) + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.2

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

$f (- \frac{1 - \sqrt{67}}{6}) = 2 (- \frac{{(1 - \sqrt{67})}^{3}}{6^{3}}) + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.3

Raise $6$ to the power of $3$ .

$f (- \frac{1 - \sqrt{67}}{6}) = 2 (- \frac{{(1 - \sqrt{67})}^{3}}{216}) + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.4

Cancel the common factor of $2$ .

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Step 11.2.1.4.1

Move the leading negative in $- \frac{{(1 - \sqrt{67})}^{3}}{216}$ into the numerator.

$f (- \frac{1 - \sqrt{67}}{6}) = 2 (\frac{- {(1 - \sqrt{67})}^{3}}{216}) + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.4.2

Factor $2$ out of $216$ .

$f (- \frac{1 - \sqrt{67}}{6}) = 2 (\frac{- {(1 - \sqrt{67})}^{3}}{2 (108)}) + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.4.3

Cancel the common factor.

$f (- \frac{1 - \sqrt{67}}{6}) = 2 (\frac{- {(1 - \sqrt{67})}^{3}}{2 \cdot 108}) + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.4.4

Rewrite the expression.

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- {(1 - \sqrt{67})}^{3}}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.5

Use the Binomial Theorem.

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1^{3} + 3 \cdot (1^{2} (- \sqrt{67})) + 3 \cdot (1 {(- \sqrt{67})}^{2}) + {(- \sqrt{67})}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6

Simplify each term.

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Step 11.2.1.6.1

One to any power is one.

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 + 3 \cdot (1^{2} (- \sqrt{67})) + 3 \cdot (1 {(- \sqrt{67})}^{2}) + {(- \sqrt{67})}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.2

One to any power is one.

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 + 3 \cdot (1 (- \sqrt{67})) + 3 \cdot (1 {(- \sqrt{67})}^{2}) + {(- \sqrt{67})}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.3

Multiply $3$ by $1$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 + 3 (- \sqrt{67}) + 3 \cdot (1 {(- \sqrt{67})}^{2}) + {(- \sqrt{67})}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.4

Multiply $- 1$ by $3$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 3 \cdot (1 {(- \sqrt{67})}^{2}) + {(- \sqrt{67})}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.5

Multiply $3$ by $1$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 3 {(- \sqrt{67})}^{2} + {(- \sqrt{67})}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.6

Apply the product rule to $- \sqrt{67}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 3 ({(- 1)}^{2} {\sqrt{67}}^{2}) + {(- \sqrt{67})}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.7

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

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 3 (1 {\sqrt{67}}^{2}) + {(- \sqrt{67})}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.8

Multiply ${\sqrt{67}}^{2}$ by $1$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 3 {\sqrt{67}}^{2} + {(- \sqrt{67})}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.9

Rewrite ${\sqrt{67}}^{2}$ as $67$ .

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Step 11.2.1.6.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{67}$ as $67^{\frac{1}{2}}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 3 {(67^{\frac{1}{2}})}^{2} + {(- \sqrt{67})}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.9.2

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

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 3 \cdot 67^{\frac{1}{2} \cdot 2} + {(- \sqrt{67})}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.9.3

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

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 3 \cdot 67^{\frac{2}{2}} + {(- \sqrt{67})}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.9.4

Cancel the common factor of $2$ .

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Step 11.2.1.6.9.4.1

Cancel the common factor.

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 3 \cdot 67^{\frac{2}{2}} + {(- \sqrt{67})}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.9.4.2

Rewrite the expression.

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 3 \cdot 67 + {(- \sqrt{67})}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.9.5

Evaluate the exponent.

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 3 \cdot 67 + {(- \sqrt{67})}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.10

Multiply $3$ by $67$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 201 + {(- \sqrt{67})}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.11

Apply the product rule to $- \sqrt{67}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 201 + {(- 1)}^{3} {\sqrt{67}}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.12

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

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 201 - {\sqrt{67}}^{3})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.13

Rewrite ${\sqrt{67}}^{3}$ as $\sqrt{67^{3}}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 201 - \sqrt{67^{3}})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.14

Raise $67$ to the power of $3$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 201 - \sqrt{300763})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.15

Rewrite $300763$ as $67^{2} \cdot 67$ .

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Step 11.2.1.6.15.1

Factor $4489$ out of $300763$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 201 - \sqrt{4489 (67)})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.15.2

Rewrite $4489$ as $67^{2}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 201 - \sqrt{67^{2} \cdot 67})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.16

Pull terms out from under the radical.

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 201 - (67 \sqrt{67}))}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.6.17

Multiply $67$ by $- 1$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (1 - 3 \sqrt{67} + 201 - 67 \sqrt{67})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.7

Add $1$ and $201$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (202 - 3 \sqrt{67} - 67 \sqrt{67})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.8

Subtract $67 \sqrt{67}$ from $- 3 \sqrt{67}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (202 - 70 \sqrt{67})}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.9

Cancel the common factor of $202 - 70 \sqrt{67}$ and $108$ .

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Step 11.2.1.9.1

Factor $2$ out of $- (202 - 70 \sqrt{67})$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{2 (- (101 - 35 \sqrt{67}))}{108} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.9.2

Cancel the common factors.

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Step 11.2.1.9.2.1

Factor $2$ out of $108$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{2 (- (101 - 35 \sqrt{67}))}{2 (54)} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.9.2.2

Cancel the common factor.

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{2 (- (101 - 35 \sqrt{67}))}{2 \cdot 54} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.9.2.3

Rewrite the expression.

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (101 - 35 \sqrt{67})}{54} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.10

Move the negative in front of the fraction.

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + {(- \frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.11

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 11.2.1.11.1

Apply the product rule to $- \frac{1 - \sqrt{67}}{6}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + {(- 1)}^{2} {(\frac{1 - \sqrt{67}}{6})}^{2} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.11.2

Apply the product rule to $\frac{1 - \sqrt{67}}{6}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + {(- 1)}^{2} (\frac{{(1 - \sqrt{67})}^{2}}{6^{2}}) - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.12

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

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + 1 (\frac{{(1 - \sqrt{67})}^{2}}{6^{2}}) - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.13

Multiply $\frac{{(1 - \sqrt{67})}^{2}}{6^{2}}$ by $1$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{{(1 - \sqrt{67})}^{2}}{6^{2}} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.14

Raise $6$ to the power of $2$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{{(1 - \sqrt{67})}^{2}}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.15

Rewrite ${(1 - \sqrt{67})}^{2}$ as $(1 - \sqrt{67}) (1 - \sqrt{67})$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{(1 - \sqrt{67}) (1 - \sqrt{67})}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.16

Expand $(1 - \sqrt{67}) (1 - \sqrt{67})$ using the FOIL Method.

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Step 11.2.1.16.1

Apply the distributive property.

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 (1 - \sqrt{67}) - \sqrt{67} (1 - \sqrt{67})}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.16.2

Apply the distributive property.

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 \cdot 1 + 1 (- \sqrt{67}) - \sqrt{67} (1 - \sqrt{67})}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.16.3

Apply the distributive property.

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 \cdot 1 + 1 (- \sqrt{67}) - \sqrt{67} \cdot 1 - \sqrt{67} (- \sqrt{67})}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.17

Simplify and combine like terms.

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Step 11.2.1.17.1

Simplify each term.

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Step 11.2.1.17.1.1

Multiply $1$ by $1$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 + 1 (- \sqrt{67}) - \sqrt{67} \cdot 1 - \sqrt{67} (- \sqrt{67})}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.17.1.2

Multiply $- \sqrt{67}$ by $1$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 - \sqrt{67} - \sqrt{67} \cdot 1 - \sqrt{67} (- \sqrt{67})}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.17.1.3

Multiply $- 1$ by $1$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 - \sqrt{67} - \sqrt{67} - \sqrt{67} (- \sqrt{67})}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.17.1.4

Multiply $- \sqrt{67} (- \sqrt{67})$ .

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Step 11.2.1.17.1.4.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 - \sqrt{67} - \sqrt{67} + 1 \sqrt{67} \sqrt{67}}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.17.1.4.2

Multiply $\sqrt{67}$ by $1$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 - \sqrt{67} - \sqrt{67} + \sqrt{67} \sqrt{67}}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.17.1.4.3

Raise $\sqrt{67}$ to the power of $1$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 - \sqrt{67} - \sqrt{67} + \sqrt{67} \sqrt{67}}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.17.1.4.4

Raise $\sqrt{67}$ to the power of $1$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 - \sqrt{67} - \sqrt{67} + \sqrt{67} \sqrt{67}}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.17.1.4.5

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

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 - \sqrt{67} - \sqrt{67} + {\sqrt{67}}^{1 + 1}}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.17.1.4.6

Add $1$ and $1$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 - \sqrt{67} - \sqrt{67} + {\sqrt{67}}^{2}}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.17.1.5

Rewrite ${\sqrt{67}}^{2}$ as $67$ .

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Step 11.2.1.17.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{67}$ as $67^{\frac{1}{2}}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 - \sqrt{67} - \sqrt{67} + {(67^{\frac{1}{2}})}^{2}}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.17.1.5.2

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

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 - \sqrt{67} - \sqrt{67} + 67^{\frac{1}{2} \cdot 2}}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.17.1.5.3

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

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 - \sqrt{67} - \sqrt{67} + 67^{\frac{2}{2}}}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.17.1.5.4

Cancel the common factor of $2$ .

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Step 11.2.1.17.1.5.4.1

Cancel the common factor.

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 - \sqrt{67} - \sqrt{67} + 67^{\frac{2}{2}}}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.17.1.5.4.2

Rewrite the expression.

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 - \sqrt{67} - \sqrt{67} + 67}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.17.1.5.5

Evaluate the exponent.

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{1 - \sqrt{67} - \sqrt{67} + 67}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.17.2

Add $1$ and $67$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{68 - \sqrt{67} - \sqrt{67}}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.17.3

Subtract $\sqrt{67}$ from $- \sqrt{67}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{68 - 2 \sqrt{67}}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.18

Cancel the common factor of $68 - 2 \sqrt{67}$ and $36$ .

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Step 11.2.1.18.1

Factor $2$ out of $68$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{2 (34) - 2 \sqrt{67}}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.18.2

Factor $2$ out of $- 2 \sqrt{67}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{2 (34) + 2 (- \sqrt{67})}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.18.3

Factor $2$ out of $2 (34) + 2 (- \sqrt{67})$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{2 (34 - \sqrt{67})}{36} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.18.4

Cancel the common factors.

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Step 11.2.1.18.4.1

Factor $2$ out of $36$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{2 (34 - \sqrt{67})}{2 \cdot 18} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.18.4.2

Cancel the common factor.

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{2 (34 - \sqrt{67})}{2 \cdot 18} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.18.4.3

Rewrite the expression.

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{34 - \sqrt{67}}{18} - 11 (- \frac{1 - \sqrt{67}}{6})$

Step 11.2.1.19

Multiply $- 11 (- \frac{1 - \sqrt{67}}{6})$ .

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Step 11.2.1.19.1

Multiply $- 1$ by $- 11$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{34 - \sqrt{67}}{18} + 11 (\frac{1 - \sqrt{67}}{6})$

Step 11.2.1.19.2

Combine $11$ and $\frac{1 - \sqrt{67}}{6}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{34 - \sqrt{67}}{18} + \frac{11 (1 - \sqrt{67})}{6}$

Step 11.2.2

Find the common denominator.

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Step 11.2.2.1

Multiply $\frac{34 - \sqrt{67}}{18}$ by $\frac{3}{3}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{34 - \sqrt{67}}{18} \cdot \frac{3}{3} + \frac{11 (1 - \sqrt{67})}{6}$

Step 11.2.2.2

Multiply $\frac{34 - \sqrt{67}}{18}$ by $\frac{3}{3}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{(34 - \sqrt{67}) \cdot 3}{18 \cdot 3} + \frac{11 (1 - \sqrt{67})}{6}$

Step 11.2.2.3

Multiply $\frac{11 (1 - \sqrt{67})}{6}$ by $\frac{9}{9}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{(34 - \sqrt{67}) \cdot 3}{18 \cdot 3} + \frac{11 (1 - \sqrt{67})}{6} \cdot \frac{9}{9}$

Step 11.2.2.4

Multiply $\frac{11 (1 - \sqrt{67})}{6}$ by $\frac{9}{9}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{(34 - \sqrt{67}) \cdot 3}{18 \cdot 3} + \frac{11 (1 - \sqrt{67}) \cdot 9}{6 \cdot 9}$

Step 11.2.2.5

Reorder the factors of $18 \cdot 3$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{(34 - \sqrt{67}) \cdot 3}{3 \cdot 18} + \frac{11 (1 - \sqrt{67}) \cdot 9}{6 \cdot 9}$

Step 11.2.2.6

Multiply $3$ by $18$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{(34 - \sqrt{67}) \cdot 3}{54} + \frac{11 (1 - \sqrt{67}) \cdot 9}{6 \cdot 9}$

Step 11.2.2.7

Multiply $6$ by $9$ .

$f (- \frac{1 - \sqrt{67}}{6}) = - \frac{101 - 35 \sqrt{67}}{54} + \frac{(34 - \sqrt{67}) \cdot 3}{54} + \frac{11 (1 - \sqrt{67}) \cdot 9}{54}$

Step 11.2.3

Combine the numerators over the common denominator.

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- (101 - 35 \sqrt{67}) + (34 - \sqrt{67}) \cdot 3 + 11 (1 - \sqrt{67}) \cdot 9}{54}$

Step 11.2.4

Simplify each term.

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Step 11.2.4.1

Apply the distributive property.

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- 1 \cdot 101 - (- 35 \sqrt{67}) + (34 - \sqrt{67}) \cdot 3 + 11 (1 - \sqrt{67}) \cdot 9}{54}$

Step 11.2.4.2

Multiply $- 1$ by $101$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- 101 - (- 35 \sqrt{67}) + (34 - \sqrt{67}) \cdot 3 + 11 (1 - \sqrt{67}) \cdot 9}{54}$

Step 11.2.4.3

Multiply $- 35$ by $- 1$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- 101 + 35 \sqrt{67} + (34 - \sqrt{67}) \cdot 3 + 11 (1 - \sqrt{67}) \cdot 9}{54}$

Step 11.2.4.4

Apply the distributive property.

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- 101 + 35 \sqrt{67} + 34 \cdot 3 - \sqrt{67} \cdot 3 + 11 (1 - \sqrt{67}) \cdot 9}{54}$

Step 11.2.4.5

Multiply $34$ by $3$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- 101 + 35 \sqrt{67} + 102 - \sqrt{67} \cdot 3 + 11 (1 - \sqrt{67}) \cdot 9}{54}$

Step 11.2.4.6

Multiply $3$ by $- 1$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- 101 + 35 \sqrt{67} + 102 - 3 \sqrt{67} + 11 (1 - \sqrt{67}) \cdot 9}{54}$

Step 11.2.4.7

Apply the distributive property.

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- 101 + 35 \sqrt{67} + 102 - 3 \sqrt{67} + (11 \cdot 1 + 11 (- \sqrt{67})) \cdot 9}{54}$

Step 11.2.4.8

Multiply $11$ by $1$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- 101 + 35 \sqrt{67} + 102 - 3 \sqrt{67} + (11 + 11 (- \sqrt{67})) \cdot 9}{54}$

Step 11.2.4.9

Multiply $- 1$ by $11$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- 101 + 35 \sqrt{67} + 102 - 3 \sqrt{67} + (11 - 11 \sqrt{67}) \cdot 9}{54}$

Step 11.2.4.10

Apply the distributive property.

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- 101 + 35 \sqrt{67} + 102 - 3 \sqrt{67} + 11 \cdot 9 - 11 \sqrt{67} \cdot 9}{54}$

Step 11.2.4.11

Multiply $11$ by $9$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- 101 + 35 \sqrt{67} + 102 - 3 \sqrt{67} + 99 - 11 \sqrt{67} \cdot 9}{54}$

Step 11.2.4.12

Multiply $9$ by $- 11$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{- 101 + 35 \sqrt{67} + 102 - 3 \sqrt{67} + 99 - 99 \sqrt{67}}{54}$

Step 11.2.5

Simplify by adding terms.

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Step 11.2.5.1

Add $- 101$ and $102$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{1 + 35 \sqrt{67} - 3 \sqrt{67} + 99 - 99 \sqrt{67}}{54}$

Step 11.2.5.2

Add $1$ and $99$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{100 + 35 \sqrt{67} - 3 \sqrt{67} - 99 \sqrt{67}}{54}$

Step 11.2.5.3

Subtract $3 \sqrt{67}$ from $35 \sqrt{67}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{100 + 32 \sqrt{67} - 99 \sqrt{67}}{54}$

Step 11.2.5.4

Subtract $99 \sqrt{67}$ from $32 \sqrt{67}$ .

$f (- \frac{1 - \sqrt{67}}{6}) = \frac{100 - 67 \sqrt{67}}{54}$

Step 11.2.6

The final answer is $\frac{100 - 67 \sqrt{67}}{54}$ .

$y = \frac{100 - 67 \sqrt{67}}{54}$

Step 12

Evaluate the second derivative at $x = - \frac{1 + \sqrt{67}}{6}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$12 (- \frac{1 + \sqrt{67}}{6}) + 2$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

Cancel the common factor of $6$ .

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Step 13.1.1.1

Move the leading negative in $- \frac{1 + \sqrt{67}}{6}$ into the numerator.

$12 \frac{- (1 + \sqrt{67})}{6} + 2$

Step 13.1.1.2

Factor $6$ out of $12$ .

$6 (2) \frac{- (1 + \sqrt{67})}{6} + 2$

Step 13.1.1.3

Cancel the common factor.

$6 \cdot 2 \frac{- (1 + \sqrt{67})}{6} + 2$

Step 13.1.1.4

Rewrite the expression.

$2 (- (1 + \sqrt{67})) + 2$

Step 13.1.2

Multiply $- 1$ by $2$ .

$- 2 (1 + \sqrt{67}) + 2$

Step 13.1.3

Apply the distributive property.

$- 2 \cdot 1 - 2 \sqrt{67} + 2$

Step 13.1.4

Multiply $- 2$ by $1$ .

$- 2 - 2 \sqrt{67} + 2$

Step 13.2

Simplify by adding numbers.

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Step 13.2.1

Add $- 2$ and $2$ .

$0 - 2 \sqrt{67}$

Step 13.2.2

Subtract $2 \sqrt{67}$ from $0$ .

$- 2 \sqrt{67}$

Step 14

$x = - \frac{1 + \sqrt{67}}{6}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - \frac{1 + \sqrt{67}}{6}$ is a local maximum

Step 15

Find the y-value when $x = - \frac{1 + \sqrt{67}}{6}$ .

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Step 15.1

Replace the variable $x$ with $- \frac{1 + \sqrt{67}}{6}$ in the expression.

$f (- \frac{1 + \sqrt{67}}{6}) = 2 {(- \frac{1 + \sqrt{67}}{6})}^{3} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 15.2.1.1.1

Apply the product rule to $- \frac{1 + \sqrt{67}}{6}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = 2 ({(- 1)}^{3} {(\frac{1 + \sqrt{67}}{6})}^{3}) + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.1.2

Apply the product rule to $\frac{1 + \sqrt{67}}{6}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = 2 ({(- 1)}^{3} (\frac{{(1 + \sqrt{67})}^{3}}{6^{3}})) + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.2

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

$f (- \frac{1 + \sqrt{67}}{6}) = 2 (- \frac{{(1 + \sqrt{67})}^{3}}{6^{3}}) + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.3

Raise $6$ to the power of $3$ .

$f (- \frac{1 + \sqrt{67}}{6}) = 2 (- \frac{{(1 + \sqrt{67})}^{3}}{216}) + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.4

Cancel the common factor of $2$ .

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Step 15.2.1.4.1

Move the leading negative in $- \frac{{(1 + \sqrt{67})}^{3}}{216}$ into the numerator.

$f (- \frac{1 + \sqrt{67}}{6}) = 2 (\frac{- {(1 + \sqrt{67})}^{3}}{216}) + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.4.2

Factor $2$ out of $216$ .

$f (- \frac{1 + \sqrt{67}}{6}) = 2 (\frac{- {(1 + \sqrt{67})}^{3}}{2 (108)}) + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.4.3

Cancel the common factor.

$f (- \frac{1 + \sqrt{67}}{6}) = 2 (\frac{- {(1 + \sqrt{67})}^{3}}{2 \cdot 108}) + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.4.4

Rewrite the expression.

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- {(1 + \sqrt{67})}^{3}}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.5

Use the Binomial Theorem.

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (1^{3} + 3 \cdot (1^{2} \sqrt{67}) + 3 \cdot (1 {\sqrt{67}}^{2}) + {\sqrt{67}}^{3})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.6

Simplify each term.

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Step 15.2.1.6.1

One to any power is one.

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (1 + 3 \cdot (1^{2} \sqrt{67}) + 3 \cdot (1 {\sqrt{67}}^{2}) + {\sqrt{67}}^{3})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.6.2

One to any power is one.

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (1 + 3 \cdot (1 \sqrt{67}) + 3 \cdot (1 {\sqrt{67}}^{2}) + {\sqrt{67}}^{3})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.6.3

Multiply $3$ by $1$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (1 + 3 \sqrt{67} + 3 \cdot (1 {\sqrt{67}}^{2}) + {\sqrt{67}}^{3})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.6.4

Multiply $3$ by $1$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (1 + 3 \sqrt{67} + 3 {\sqrt{67}}^{2} + {\sqrt{67}}^{3})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.6.5

Rewrite ${\sqrt{67}}^{2}$ as $67$ .

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Step 15.2.1.6.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{67}$ as $67^{\frac{1}{2}}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (1 + 3 \sqrt{67} + 3 {(67^{\frac{1}{2}})}^{2} + {\sqrt{67}}^{3})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.6.5.2

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

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (1 + 3 \sqrt{67} + 3 \cdot 67^{\frac{1}{2} \cdot 2} + {\sqrt{67}}^{3})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.6.5.3

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

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (1 + 3 \sqrt{67} + 3 \cdot 67^{\frac{2}{2}} + {\sqrt{67}}^{3})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.6.5.4

Cancel the common factor of $2$ .

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Step 15.2.1.6.5.4.1

Cancel the common factor.

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (1 + 3 \sqrt{67} + 3 \cdot 67^{\frac{2}{2}} + {\sqrt{67}}^{3})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.6.5.4.2

Rewrite the expression.

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (1 + 3 \sqrt{67} + 3 \cdot 67 + {\sqrt{67}}^{3})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.6.5.5

Evaluate the exponent.

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (1 + 3 \sqrt{67} + 3 \cdot 67 + {\sqrt{67}}^{3})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.6.6

Multiply $3$ by $67$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (1 + 3 \sqrt{67} + 201 + {\sqrt{67}}^{3})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.6.7

Rewrite ${\sqrt{67}}^{3}$ as $\sqrt{67^{3}}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (1 + 3 \sqrt{67} + 201 + \sqrt{67^{3}})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.6.8

Raise $67$ to the power of $3$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (1 + 3 \sqrt{67} + 201 + \sqrt{300763})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.6.9

Rewrite $300763$ as $67^{2} \cdot 67$ .

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Step 15.2.1.6.9.1

Factor $4489$ out of $300763$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (1 + 3 \sqrt{67} + 201 + \sqrt{4489 (67)})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.6.9.2

Rewrite $4489$ as $67^{2}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (1 + 3 \sqrt{67} + 201 + \sqrt{67^{2} \cdot 67})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.6.10

Pull terms out from under the radical.

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (1 + 3 \sqrt{67} + 201 + 67 \sqrt{67})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.7

Add $1$ and $201$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (202 + 3 \sqrt{67} + 67 \sqrt{67})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.8

Add $3 \sqrt{67}$ and $67 \sqrt{67}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (202 + 70 \sqrt{67})}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.9

Cancel the common factor of $202 + 70 \sqrt{67}$ and $108$ .

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Step 15.2.1.9.1

Factor $2$ out of $- (202 + 70 \sqrt{67})$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{2 (- (101 + 35 \sqrt{67}))}{108} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.9.2

Cancel the common factors.

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Step 15.2.1.9.2.1

Factor $2$ out of $108$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{2 (- (101 + 35 \sqrt{67}))}{2 (54)} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.9.2.2

Cancel the common factor.

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{2 (- (101 + 35 \sqrt{67}))}{2 \cdot 54} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.9.2.3

Rewrite the expression.

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (101 + 35 \sqrt{67})}{54} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.10

Move the negative in front of the fraction.

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + {(- \frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.11

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 15.2.1.11.1

Apply the product rule to $- \frac{1 + \sqrt{67}}{6}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + {(- 1)}^{2} {(\frac{1 + \sqrt{67}}{6})}^{2} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.11.2

Apply the product rule to $\frac{1 + \sqrt{67}}{6}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + {(- 1)}^{2} (\frac{{(1 + \sqrt{67})}^{2}}{6^{2}}) - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.12

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

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + 1 (\frac{{(1 + \sqrt{67})}^{2}}{6^{2}}) - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.13

Multiply $\frac{{(1 + \sqrt{67})}^{2}}{6^{2}}$ by $1$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{{(1 + \sqrt{67})}^{2}}{6^{2}} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.14

Raise $6$ to the power of $2$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{{(1 + \sqrt{67})}^{2}}{36} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.15

Rewrite ${(1 + \sqrt{67})}^{2}$ as $(1 + \sqrt{67}) (1 + \sqrt{67})$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{(1 + \sqrt{67}) (1 + \sqrt{67})}{36} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.16

Expand $(1 + \sqrt{67}) (1 + \sqrt{67})$ using the FOIL Method.

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Step 15.2.1.16.1

Apply the distributive property.

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{1 (1 + \sqrt{67}) + \sqrt{67} (1 + \sqrt{67})}{36} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.16.2

Apply the distributive property.

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{1 \cdot 1 + 1 \sqrt{67} + \sqrt{67} (1 + \sqrt{67})}{36} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.16.3

Apply the distributive property.

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{1 \cdot 1 + 1 \sqrt{67} + \sqrt{67} \cdot 1 + \sqrt{67} \sqrt{67}}{36} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.17

Simplify and combine like terms.

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Step 15.2.1.17.1

Simplify each term.

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Step 15.2.1.17.1.1

Multiply $1$ by $1$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{1 + 1 \sqrt{67} + \sqrt{67} \cdot 1 + \sqrt{67} \sqrt{67}}{36} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.17.1.2

Multiply $\sqrt{67}$ by $1$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{1 + \sqrt{67} + \sqrt{67} \cdot 1 + \sqrt{67} \sqrt{67}}{36} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.17.1.3

Multiply $\sqrt{67}$ by $1$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{1 + \sqrt{67} + \sqrt{67} + \sqrt{67} \sqrt{67}}{36} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.17.1.4

Combine using the product rule for radicals.

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{1 + \sqrt{67} + \sqrt{67} + \sqrt{67 \cdot 67}}{36} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.17.1.5

Multiply $67$ by $67$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{1 + \sqrt{67} + \sqrt{67} + \sqrt{4489}}{36} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.17.1.6

Rewrite $4489$ as $67^{2}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{1 + \sqrt{67} + \sqrt{67} + \sqrt{67^{2}}}{36} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.17.1.7

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

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{1 + \sqrt{67} + \sqrt{67} + 67}{36} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.17.2

Add $1$ and $67$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{68 + \sqrt{67} + \sqrt{67}}{36} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.17.3

Add $\sqrt{67}$ and $\sqrt{67}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{68 + 2 \sqrt{67}}{36} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.18

Cancel the common factor of $68 + 2 \sqrt{67}$ and $36$ .

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Step 15.2.1.18.1

Factor $2$ out of $68$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{2 \cdot 34 + 2 \sqrt{67}}{36} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.18.2

Factor $2$ out of $2 \sqrt{67}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{2 \cdot 34 + 2 (\sqrt{67})}{36} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.18.3

Factor $2$ out of $2 \cdot 34 + 2 (\sqrt{67})$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{2 \cdot (34 + \sqrt{67})}{36} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.18.4

Cancel the common factors.

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Step 15.2.1.18.4.1

Factor $2$ out of $36$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{2 \cdot (34 + \sqrt{67})}{2 (18)} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.18.4.2

Cancel the common factor.

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{2 \cdot (34 + \sqrt{67})}{2 \cdot 18} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.18.4.3

Rewrite the expression.

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{34 + \sqrt{67}}{18} - 11 (- \frac{1 + \sqrt{67}}{6})$

Step 15.2.1.19

Multiply $- 11 (- \frac{1 + \sqrt{67}}{6})$ .

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Step 15.2.1.19.1

Multiply $- 1$ by $- 11$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{34 + \sqrt{67}}{18} + 11 (\frac{1 + \sqrt{67}}{6})$

Step 15.2.1.19.2

Combine $11$ and $\frac{1 + \sqrt{67}}{6}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{34 + \sqrt{67}}{18} + \frac{11 (1 + \sqrt{67})}{6}$

Step 15.2.2

Find the common denominator.

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Step 15.2.2.1

Multiply $\frac{34 + \sqrt{67}}{18}$ by $\frac{3}{3}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{34 + \sqrt{67}}{18} \cdot \frac{3}{3} + \frac{11 (1 + \sqrt{67})}{6}$

Step 15.2.2.2

Multiply $\frac{34 + \sqrt{67}}{18}$ by $\frac{3}{3}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{(34 + \sqrt{67}) \cdot 3}{18 \cdot 3} + \frac{11 (1 + \sqrt{67})}{6}$

Step 15.2.2.3

Multiply $\frac{11 (1 + \sqrt{67})}{6}$ by $\frac{9}{9}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{(34 + \sqrt{67}) \cdot 3}{18 \cdot 3} + \frac{11 (1 + \sqrt{67})}{6} \cdot \frac{9}{9}$

Step 15.2.2.4

Multiply $\frac{11 (1 + \sqrt{67})}{6}$ by $\frac{9}{9}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{(34 + \sqrt{67}) \cdot 3}{18 \cdot 3} + \frac{11 (1 + \sqrt{67}) \cdot 9}{6 \cdot 9}$

Step 15.2.2.5

Reorder the factors of $18 \cdot 3$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{(34 + \sqrt{67}) \cdot 3}{3 \cdot 18} + \frac{11 (1 + \sqrt{67}) \cdot 9}{6 \cdot 9}$

Step 15.2.2.6

Multiply $3$ by $18$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{(34 + \sqrt{67}) \cdot 3}{54} + \frac{11 (1 + \sqrt{67}) \cdot 9}{6 \cdot 9}$

Step 15.2.2.7

Multiply $6$ by $9$ .

$f (- \frac{1 + \sqrt{67}}{6}) = - \frac{101 + 35 \sqrt{67}}{54} + \frac{(34 + \sqrt{67}) \cdot 3}{54} + \frac{11 (1 + \sqrt{67}) \cdot 9}{54}$

Step 15.2.3

Combine the numerators over the common denominator.

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- (101 + 35 \sqrt{67}) + (34 + \sqrt{67}) \cdot 3 + 11 (1 + \sqrt{67}) \cdot 9}{54}$

Step 15.2.4

Simplify each term.

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Step 15.2.4.1

Apply the distributive property.

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- 1 \cdot 101 - (35 \sqrt{67}) + (34 + \sqrt{67}) \cdot 3 + 11 (1 + \sqrt{67}) \cdot 9}{54}$

Step 15.2.4.2

Multiply $- 1$ by $101$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- 101 - (35 \sqrt{67}) + (34 + \sqrt{67}) \cdot 3 + 11 (1 + \sqrt{67}) \cdot 9}{54}$

Step 15.2.4.3

Multiply $35$ by $- 1$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- 101 - 35 \sqrt{67} + (34 + \sqrt{67}) \cdot 3 + 11 (1 + \sqrt{67}) \cdot 9}{54}$

Step 15.2.4.4

Apply the distributive property.

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- 101 - 35 \sqrt{67} + 34 \cdot 3 + \sqrt{67} \cdot 3 + 11 (1 + \sqrt{67}) \cdot 9}{54}$

Step 15.2.4.5

Multiply $34$ by $3$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- 101 - 35 \sqrt{67} + 102 + \sqrt{67} \cdot 3 + 11 (1 + \sqrt{67}) \cdot 9}{54}$

Step 15.2.4.6

Move $3$ to the left of $\sqrt{67}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- 101 - 35 \sqrt{67} + 102 + 3 \cdot \sqrt{67} + 11 (1 + \sqrt{67}) \cdot 9}{54}$

Step 15.2.4.7

Apply the distributive property.

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- 101 - 35 \sqrt{67} + 102 + 3 \sqrt{67} + (11 \cdot 1 + 11 \sqrt{67}) \cdot 9}{54}$

Step 15.2.4.8

Multiply $11$ by $1$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- 101 - 35 \sqrt{67} + 102 + 3 \sqrt{67} + (11 + 11 \sqrt{67}) \cdot 9}{54}$

Step 15.2.4.9

Apply the distributive property.

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- 101 - 35 \sqrt{67} + 102 + 3 \sqrt{67} + 11 \cdot 9 + 11 \sqrt{67} \cdot 9}{54}$

Step 15.2.4.10

Multiply $11$ by $9$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- 101 - 35 \sqrt{67} + 102 + 3 \sqrt{67} + 99 + 11 \sqrt{67} \cdot 9}{54}$

Step 15.2.4.11

Multiply $9$ by $11$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{- 101 - 35 \sqrt{67} + 102 + 3 \sqrt{67} + 99 + 99 \sqrt{67}}{54}$

Step 15.2.5

Simplify by adding terms.

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Step 15.2.5.1

Add $- 101$ and $102$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{1 - 35 \sqrt{67} + 3 \sqrt{67} + 99 + 99 \sqrt{67}}{54}$

Step 15.2.5.2

Add $1$ and $99$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{100 - 35 \sqrt{67} + 3 \sqrt{67} + 99 \sqrt{67}}{54}$

Step 15.2.5.3

Add $- 35 \sqrt{67}$ and $3 \sqrt{67}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{100 - 32 \sqrt{67} + 99 \sqrt{67}}{54}$

Step 15.2.5.4

Add $- 32 \sqrt{67}$ and $99 \sqrt{67}$ .

$f (- \frac{1 + \sqrt{67}}{6}) = \frac{100 + 67 \sqrt{67}}{54}$

Step 15.2.6

The final answer is $\frac{100 + 67 \sqrt{67}}{54}$ .

$y = \frac{100 + 67 \sqrt{67}}{54}$

Step 16

These are the local extrema for $f (x) = 2 x^{3} + x^{2} - 11 x$ .

$(- \frac{1 - \sqrt{67}}{6}, \frac{100 - 67 \sqrt{67}}{54})$ is a local minima

$(- \frac{1 + \sqrt{67}}{6}, \frac{100 + 67 \sqrt{67}}{54})$ is a local maxima

Step 17