Calculus Examples

$f (x) = 64 x^{2} + \frac{54}{x} - 3$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $64 x^{2} + \frac{54}{x} - 3$ with respect to $x$ is $\frac{d}{d x} [64 x^{2}] + \frac{d}{d x} [\frac{54}{x}] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [64 x^{2}] + \frac{d}{d x} [\frac{54}{x}] + \frac{d}{d x} [- 3]$

Step 1.1.2

Evaluate $\frac{d}{d x} [64 x^{2}]$ .

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Step 1.1.2.1

Since $64$ is constant with respect to $x$ , the derivative of $64 x^{2}$ with respect to $x$ is $64 \frac{d}{d x} [x^{2}]$ .

$64 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{54}{x}] + \frac{d}{d x} [- 3]$

Step 1.1.2.2

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

$64 (2 x) + \frac{d}{d x} [\frac{54}{x}] + \frac{d}{d x} [- 3]$

Step 1.1.2.3

Multiply $2$ by $64$ .

$128 x + \frac{d}{d x} [\frac{54}{x}] + \frac{d}{d x} [- 3]$

Step 1.1.3

Evaluate $\frac{d}{d x} [\frac{54}{x}]$ .

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Step 1.1.3.1

Since $54$ is constant with respect to $x$ , the derivative of $\frac{54}{x}$ with respect to $x$ is $54 \frac{d}{d x} [\frac{1}{x}]$ .

$128 x + 54 \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [- 3]$

Step 1.1.3.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$128 x + 54 \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [- 3]$

Step 1.1.3.3

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

$128 x + 54 (- x^{- 2}) + \frac{d}{d x} [- 3]$

Step 1.1.3.4

Multiply $- 1$ by $54$ .

$128 x - 54 x^{- 2} + \frac{d}{d x} [- 3]$

Step 1.1.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$128 x - 54 x^{- 2} + 0$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$128 x - 54 \frac{1}{x^{2}} + 0$

Step 1.1.5.2

Combine terms.

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Step 1.1.5.2.1

Combine $- 54$ and $\frac{1}{x^{2}}$ .

$128 x + \frac{- 54}{x^{2}} + 0$

Step 1.1.5.2.2

Move the negative in front of the fraction.

$128 x - \frac{54}{x^{2}} + 0$

Step 1.1.5.2.3

Add $128 x - \frac{54}{x^{2}}$ and $0$ .

$f' (x) = 128 x - \frac{54}{x^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $128 x - \frac{54}{x^{2}}$ .

$128 x - \frac{54}{x^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $128 x - \frac{54}{x^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$128 x - \frac{54}{x^{2}} = 0$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x^{2}, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 2.3

Multiply each term in $128 x - \frac{54}{x^{2}} = 0$ by $x^{2}$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $128 x - \frac{54}{x^{2}} = 0$ by $x^{2}$ .

$128 x \cdot x^{2} - \frac{54}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 2.3.2.1.1.1

Move $x^{2}$ .

$128 (x^{2} x) - \frac{54}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.1.2

Multiply $x^{2}$ by $x$ .

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Step 2.3.2.1.1.2.1

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

$128 (x^{2} x^{1}) - \frac{54}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.1.2.2

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

$128 x^{2 + 1} - \frac{54}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.1.3

Add $2$ and $1$ .

$128 x^{3} - \frac{54}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.2

Cancel the common factor of $x^{2}$ .

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Step 2.3.2.1.2.1

Move the leading negative in $- \frac{54}{x^{2}}$ into the numerator.

$128 x^{3} + \frac{- 54}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.2.2

Cancel the common factor.

$128 x^{3} + \frac{- 54}{x^{2}} x^{2} = 0 x^{2}$

Step 2.3.2.1.2.3

Rewrite the expression.

$128 x^{3} - 54 = 0 x^{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0$ by $x^{2}$ .

$128 x^{3} - 54 = 0$

Step 2.4

Solve the equation.

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Step 2.4.1

Add $54$ to both sides of the equation.

$128 x^{3} = 54$

Step 2.4.2

Subtract $54$ from both sides of the equation.

$128 x^{3} - 54 = 0$

Step 2.4.3

Factor the left side of the equation.

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Step 2.4.3.1

Factor $2$ out of $128 x^{3} - 54$ .

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Step 2.4.3.1.1

Factor $2$ out of $128 x^{3}$ .

$2 (64 x^{3}) - 54 = 0$

Step 2.4.3.1.2

Factor $2$ out of $- 54$ .

$2 (64 x^{3}) + 2 (- 27) = 0$

Step 2.4.3.1.3

Factor $2$ out of $2 (64 x^{3}) + 2 (- 27)$ .

$2 (64 x^{3} - 27) = 0$

Step 2.4.3.2

Rewrite $64 x^{3}$ as ${(4 x)}^{3}$ .

$2 ({(4 x)}^{3} - 27) = 0$

Step 2.4.3.3

Rewrite $27$ as $3^{3}$ .

$2 ({(4 x)}^{3} - 3^{3}) = 0$

Step 2.4.3.4

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 4 x$ and $b = 3$ .

$2 ((4 x - 3) ({(4 x)}^{2} + 4 x \cdot 3 + 3^{2})) = 0$

Step 2.4.3.5

Factor.

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Step 2.4.3.5.1

Simplify.

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Step 2.4.3.5.1.1

Apply the product rule to $4 x$ .

$2 ((4 x - 3) (4^{2} x^{2} + 4 x \cdot 3 + 3^{2})) = 0$

Step 2.4.3.5.1.2

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

$2 ((4 x - 3) (16 x^{2} + 4 x \cdot 3 + 3^{2})) = 0$

Step 2.4.3.5.1.3

Multiply $3$ by $4$ .

$2 ((4 x - 3) (16 x^{2} + 12 x + 3^{2})) = 0$

Step 2.4.3.5.1.4

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

$2 ((4 x - 3) (16 x^{2} + 12 x + 9)) = 0$

Step 2.4.3.5.2

Remove unnecessary parentheses.

$2 (4 x - 3) (16 x^{2} + 12 x + 9) = 0$

Step 2.4.4

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

$4 x - 3 = 0$

$16 x^{2} + 12 x + 9 = 0$

Step 2.4.5

Set $4 x - 3$ equal to $0$ and solve for $x$ .

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Step 2.4.5.1

Set $4 x - 3$ equal to $0$ .

$4 x - 3 = 0$

Step 2.4.5.2

Solve $4 x - 3 = 0$ for $x$ .

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Step 2.4.5.2.1

Add $3$ to both sides of the equation.

$4 x = 3$

Step 2.4.5.2.2

Divide each term in $4 x = 3$ by $4$ and simplify.

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Step 2.4.5.2.2.1

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Step 2.4.5.2.2.2

Simplify the left side.

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Step 2.4.5.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.4.5.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Step 2.4.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{4}$

Step 2.4.6

Set $16 x^{2} + 12 x + 9$ equal to $0$ and solve for $x$ .

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Step 2.4.6.1

Set $16 x^{2} + 12 x + 9$ equal to $0$ .

$16 x^{2} + 12 x + 9 = 0$

Step 2.4.6.2

Solve $16 x^{2} + 12 x + 9 = 0$ for $x$ .

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Step 2.4.6.2.1

Use the quadratic formula to find the solutions.

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

Step 2.4.6.2.2

Substitute the values $a = 16$ , $b = 12$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (16 \cdot 9)}}{2 \cdot 16}$

Step 2.4.6.2.3

Simplify.

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Step 2.4.6.2.3.1

Simplify the numerator.

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Step 2.4.6.2.3.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 16 \cdot 9}}{2 \cdot 16}$

Step 2.4.6.2.3.1.2

Multiply $- 4 \cdot 16 \cdot 9$ .

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Step 2.4.6.2.3.1.2.1

Multiply $- 4$ by $16$ .

$x = \frac{- 12 \pm \sqrt{144 - 64 \cdot 9}}{2 \cdot 16}$

Step 2.4.6.2.3.1.2.2

Multiply $- 64$ by $9$ .

$x = \frac{- 12 \pm \sqrt{144 - 576}}{2 \cdot 16}$

Step 2.4.6.2.3.1.3

Subtract $576$ from $144$ .

$x = \frac{- 12 \pm \sqrt{- 432}}{2 \cdot 16}$

Step 2.4.6.2.3.1.4

Rewrite $- 432$ as $- 1 (432)$ .

$x = \frac{- 12 \pm \sqrt{- 1 \cdot 432}}{2 \cdot 16}$

Step 2.4.6.2.3.1.5

Rewrite $\sqrt{- 1 (432)}$ as $\sqrt{- 1} \cdot \sqrt{432}$ .

$x = \frac{- 12 \pm \sqrt{- 1} \cdot \sqrt{432}}{2 \cdot 16}$

Step 2.4.6.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 12 \pm i \cdot \sqrt{432}}{2 \cdot 16}$

Step 2.4.6.2.3.1.7

Rewrite $432$ as $12^{2} \cdot 3$ .

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Step 2.4.6.2.3.1.7.1

Factor $144$ out of $432$ .

$x = \frac{- 12 \pm i \cdot \sqrt{144 (3)}}{2 \cdot 16}$

Step 2.4.6.2.3.1.7.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 12 \pm i \cdot \sqrt{12^{2} \cdot 3}}{2 \cdot 16}$

Step 2.4.6.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 12 \pm i \cdot (12 \sqrt{3})}{2 \cdot 16}$

Step 2.4.6.2.3.1.9

Move $12$ to the left of $i$ .

$x = \frac{- 12 \pm 12 i \sqrt{3}}{2 \cdot 16}$

Step 2.4.6.2.3.2

Multiply $2$ by $16$ .

$x = \frac{- 12 \pm 12 i \sqrt{3}}{32}$

Step 2.4.6.2.3.3

Simplify $\frac{- 12 \pm 12 i \sqrt{3}}{32}$ .

$x = \frac{- 3 \pm 3 i \sqrt{3}}{8}$

Step 2.4.6.2.4

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

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Step 2.4.6.2.4.1

Simplify the numerator.

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Step 2.4.6.2.4.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 16 \cdot 9}}{2 \cdot 16}$

Step 2.4.6.2.4.1.2

Multiply $- 4 \cdot 16 \cdot 9$ .

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Step 2.4.6.2.4.1.2.1

Multiply $- 4$ by $16$ .

$x = \frac{- 12 \pm \sqrt{144 - 64 \cdot 9}}{2 \cdot 16}$

Step 2.4.6.2.4.1.2.2

Multiply $- 64$ by $9$ .

$x = \frac{- 12 \pm \sqrt{144 - 576}}{2 \cdot 16}$

Step 2.4.6.2.4.1.3

Subtract $576$ from $144$ .

$x = \frac{- 12 \pm \sqrt{- 432}}{2 \cdot 16}$

Step 2.4.6.2.4.1.4

Rewrite $- 432$ as $- 1 (432)$ .

$x = \frac{- 12 \pm \sqrt{- 1 \cdot 432}}{2 \cdot 16}$

Step 2.4.6.2.4.1.5

Rewrite $\sqrt{- 1 (432)}$ as $\sqrt{- 1} \cdot \sqrt{432}$ .

$x = \frac{- 12 \pm \sqrt{- 1} \cdot \sqrt{432}}{2 \cdot 16}$

Step 2.4.6.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 12 \pm i \cdot \sqrt{432}}{2 \cdot 16}$

Step 2.4.6.2.4.1.7

Rewrite $432$ as $12^{2} \cdot 3$ .

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Step 2.4.6.2.4.1.7.1

Factor $144$ out of $432$ .

$x = \frac{- 12 \pm i \cdot \sqrt{144 (3)}}{2 \cdot 16}$

Step 2.4.6.2.4.1.7.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 12 \pm i \cdot \sqrt{12^{2} \cdot 3}}{2 \cdot 16}$

Step 2.4.6.2.4.1.8

Pull terms out from under the radical.

$x = \frac{- 12 \pm i \cdot (12 \sqrt{3})}{2 \cdot 16}$

Step 2.4.6.2.4.1.9

Move $12$ to the left of $i$ .

$x = \frac{- 12 \pm 12 i \sqrt{3}}{2 \cdot 16}$

Step 2.4.6.2.4.2

Multiply $2$ by $16$ .

$x = \frac{- 12 \pm 12 i \sqrt{3}}{32}$

Step 2.4.6.2.4.3

Simplify $\frac{- 12 \pm 12 i \sqrt{3}}{32}$ .

$x = \frac{- 3 \pm 3 i \sqrt{3}}{8}$

Step 2.4.6.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{- 3 + 3 i \sqrt{3}}{8}$

Step 2.4.6.2.4.5

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + 3 i \sqrt{3}}{8}$

Step 2.4.6.2.4.6

Factor $- 1$ out of $3 i \sqrt{3}$ .

$x = \frac{- 1 \cdot 3 - (- 3 i \sqrt{3})}{8}$

Step 2.4.6.2.4.7

Factor $- 1$ out of $- 1 (3) - (- 3 i \sqrt{3})$ .

$x = \frac{- 1 (3 - 3 i \sqrt{3})}{8}$

Step 2.4.6.2.4.8

Move the negative in front of the fraction.

$x = - \frac{3 - 3 i \sqrt{3}}{8}$

Step 2.4.6.2.5

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

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Step 2.4.6.2.5.1

Simplify the numerator.

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Step 2.4.6.2.5.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 16 \cdot 9}}{2 \cdot 16}$

Step 2.4.6.2.5.1.2

Multiply $- 4 \cdot 16 \cdot 9$ .

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Step 2.4.6.2.5.1.2.1

Multiply $- 4$ by $16$ .

$x = \frac{- 12 \pm \sqrt{144 - 64 \cdot 9}}{2 \cdot 16}$

Step 2.4.6.2.5.1.2.2

Multiply $- 64$ by $9$ .

$x = \frac{- 12 \pm \sqrt{144 - 576}}{2 \cdot 16}$

Step 2.4.6.2.5.1.3

Subtract $576$ from $144$ .

$x = \frac{- 12 \pm \sqrt{- 432}}{2 \cdot 16}$

Step 2.4.6.2.5.1.4

Rewrite $- 432$ as $- 1 (432)$ .

$x = \frac{- 12 \pm \sqrt{- 1 \cdot 432}}{2 \cdot 16}$

Step 2.4.6.2.5.1.5

Rewrite $\sqrt{- 1 (432)}$ as $\sqrt{- 1} \cdot \sqrt{432}$ .

$x = \frac{- 12 \pm \sqrt{- 1} \cdot \sqrt{432}}{2 \cdot 16}$

Step 2.4.6.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 12 \pm i \cdot \sqrt{432}}{2 \cdot 16}$

Step 2.4.6.2.5.1.7

Rewrite $432$ as $12^{2} \cdot 3$ .

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Step 2.4.6.2.5.1.7.1

Factor $144$ out of $432$ .

$x = \frac{- 12 \pm i \cdot \sqrt{144 (3)}}{2 \cdot 16}$

Step 2.4.6.2.5.1.7.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 12 \pm i \cdot \sqrt{12^{2} \cdot 3}}{2 \cdot 16}$

Step 2.4.6.2.5.1.8

Pull terms out from under the radical.

$x = \frac{- 12 \pm i \cdot (12 \sqrt{3})}{2 \cdot 16}$

Step 2.4.6.2.5.1.9

Move $12$ to the left of $i$ .

$x = \frac{- 12 \pm 12 i \sqrt{3}}{2 \cdot 16}$

Step 2.4.6.2.5.2

Multiply $2$ by $16$ .

$x = \frac{- 12 \pm 12 i \sqrt{3}}{32}$

Step 2.4.6.2.5.3

Simplify $\frac{- 12 \pm 12 i \sqrt{3}}{32}$ .

$x = \frac{- 3 \pm 3 i \sqrt{3}}{8}$

Step 2.4.6.2.5.4

Change the $\pm$ to $-$ .

$x = \frac{- 3 - 3 i \sqrt{3}}{8}$

Step 2.4.6.2.5.5

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - 3 i \sqrt{3}}{8}$

Step 2.4.6.2.5.6

Factor $- 1$ out of $- 3 i \sqrt{3}$ .

$x = \frac{- 1 \cdot 3 - (3 i \sqrt{3})}{8}$

Step 2.4.6.2.5.7

Factor $- 1$ out of $- 1 (3) - (3 i \sqrt{3})$ .

$x = \frac{- 1 (3 + 3 i \sqrt{3})}{8}$

Step 2.4.6.2.5.8

Move the negative in front of the fraction.

$x = - \frac{3 + 3 i \sqrt{3}}{8}$

Step 2.4.6.2.6

The final answer is the combination of both solutions.

$x = - \frac{3 - 3 i \sqrt{3}}{8}, - \frac{3 + 3 i \sqrt{3}}{8}$

Step 2.4.7

The final solution is all the values that make $2 (4 x - 3) (16 x^{2} + 12 x + 9) = 0$ true.

$x = \frac{3}{4}, - \frac{3 - 3 i \sqrt{3}}{8}, - \frac{3 + 3 i \sqrt{3}}{8}$

Step 3

The values which make the derivative equal to $0$ are $\frac{3}{4}$ .

$\frac{3}{4}$

Step 4

Find where the derivative is undefined.

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Step 4.1

Set the denominator in $\frac{54}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 4.2

Solve for $x$ .

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Step 4.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 4.2.2.2

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

$x = \pm 0$

Step 4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, 0) \cup (0, \frac{3}{4}) \cup (\frac{3}{4}, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

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

$f' (- 1) = 128 (- 1) - \frac{54}{{(- 1)}^{2}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $128$ by $- 1$ .

$f' (- 1) = - 128 - \frac{54}{{(- 1)}^{2}}$

Step 6.2.1.2

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

$f' (- 1) = - 128 - \frac{54}{1}$

Step 6.2.1.3

Divide $54$ by $1$ .

$f' (- 1) = - 128 - 1 \cdot 54$

Step 6.2.1.4

Multiply $- 1$ by $54$ .

$f' (- 1) = - 128 - 54$

Step 6.2.2

Subtract $54$ from $- 128$ .

$f' (- 1) = - 182$

Step 6.2.3

The final answer is $- 182$ .

$- 182$

Step 6.3

At $x = - 1$ the derivative is $- 182$ . Since this is negative, the function is decreasing on $(- \infty, 0)$ .

Decreasing on $(- \infty, 0)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(0, \frac{3}{4})$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0.375$ in the expression.

$f' (0.375) = 128 (0.375) - \frac{54}{{(0.375)}^{2}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Multiply $128$ by $0.375$ .

$f' (0.375) = 48 - \frac{54}{{(0.375)}^{2}}$

Step 7.2.1.2

Raise $0.375$ to the power of $2$ .

$f' (0.375) = 48 - \frac{54}{0.140625}$

Step 7.2.1.3

Divide $54$ by $0.140625$ .

$f' (0.375) = 48 - 1 \cdot 384$

Step 7.2.1.4

Multiply $- 1$ by $384$ .

$f' (0.375) = 48 - 384$

Step 7.2.2

Subtract $384$ from $48$ .

$f' (0.375) = - 336$

Step 7.2.3

The final answer is $- 336$ .

$- 336$

Step 7.3

At $x = 0.375$ the derivative is $- 336$ . Since this is negative, the function is decreasing on $(0, \frac{3}{4})$ .

Decreasing on $(0, \frac{3}{4})$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(\frac{3}{4}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $1.75$ in the expression.

$f' (1.75) = 128 (1.75) - \frac{54}{{(1.75)}^{2}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Multiply $128$ by $1.75$ .

$f' (1.75) = 224 - \frac{54}{{(1.75)}^{2}}$

Step 8.2.1.2

Raise $1.75$ to the power of $2$ .

$f' (1.75) = 224 - \frac{54}{3.0625}$

Step 8.2.1.3

Divide $54$ by $3.0625$ .

$f' (1.75) = 224 - 1 \cdot 17.63265306$

Step 8.2.1.4

Multiply $- 1$ by $17.63265306$ .

$f' (1.75) = 224 - 17.63265306$

Step 8.2.2

Subtract $17.63265306$ from $224$ .

$f' (1.75) = 206.36734693$

Step 8.2.3

The final answer is $206.36734693$ .

$206.36734693$

Step 8.3

At $x = 1.75$ the derivative is $206.36734693$ . Since this is positive, the function is increasing on $(\frac{3}{4}, \infty)$ .

Increasing on $(\frac{3}{4}, \infty)$ since $f' (x) > 0$

Step 9

List the intervals on which the function is increasing and decreasing.

Increasing on: $(\frac{3}{4}, \infty)$

Decreasing on: $(- \infty, 0), (0, \frac{3}{4})$

Step 10