Calculus Examples

$f (x) = \frac{3}{32} x^{2} - 4 x^{- 2}$

Step 1

Find the second 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 $\frac{3}{32} x^{2} - 4 x^{- 2}$ with respect to $x$ is $\frac{d}{d x} [\frac{3}{32} x^{2}] + \frac{d}{d x} [- 4 x^{- 2}]$ .

$\frac{d}{d x} [\frac{3}{32} x^{2}] + \frac{d}{d x} [- 4 x^{- 2}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{3}{32} x^{2}]$ .

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Step 1.1.2.1

Since $\frac{3}{32}$ is constant with respect to $x$ , the derivative of $\frac{3}{32} x^{2}$ with respect to $x$ is $\frac{3}{32} \frac{d}{d x} [x^{2}]$ .

$\frac{3}{32} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x^{- 2}]$

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 $n = 2$ .

$\frac{3}{32} (2 x) + \frac{d}{d x} [- 4 x^{- 2}]$

Step 1.1.2.3

Combine $2$ and $\frac{3}{32}$ .

$\frac{2 \cdot 3}{32} x + \frac{d}{d x} [- 4 x^{- 2}]$

Step 1.1.2.4

Multiply $2$ by $3$ .

$\frac{6}{32} x + \frac{d}{d x} [- 4 x^{- 2}]$

Step 1.1.2.5

Combine $\frac{6}{32}$ and $x$ .

$\frac{6 x}{32} + \frac{d}{d x} [- 4 x^{- 2}]$

Step 1.1.2.6

Cancel the common factor of $6$ and $32$ .

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Step 1.1.2.6.1

Factor $2$ out of $6 x$ .

$\frac{2 (3 x)}{32} + \frac{d}{d x} [- 4 x^{- 2}]$

Step 1.1.2.6.2

Cancel the common factors.

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Step 1.1.2.6.2.1

Factor $2$ out of $32$ .

$\frac{2 (3 x)}{2 (16)} + \frac{d}{d x} [- 4 x^{- 2}]$

Step 1.1.2.6.2.2

Cancel the common factor.

$\frac{2 (3 x)}{2 \cdot 16} + \frac{d}{d x} [- 4 x^{- 2}]$

Step 1.1.2.6.2.3

Rewrite the expression.

$\frac{3 x}{16} + \frac{d}{d x} [- 4 x^{- 2}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 4 x^{- 2}]$ .

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Step 1.1.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{- 2}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{- 2}]$ .

$\frac{3 x}{16} - 4 \frac{d}{d x} [x^{- 2}]$

Step 1.1.3.2

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

$\frac{3 x}{16} - 4 (- 2 x^{- 3})$

Step 1.1.3.3

Multiply $- 2$ by $- 4$ .

$\frac{3 x}{16} + 8 x^{- 3}$

Step 1.1.4

Simplify.

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Step 1.1.4.1

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

$\frac{3 x}{16} + 8 \frac{1}{x^{3}}$

Step 1.1.4.2

Combine $8$ and $\frac{1}{x^{3}}$ .

$f' (x) = \frac{3 x}{16} + \frac{8}{x^{3}}$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $\frac{3 x}{16} + \frac{8}{x^{3}}$ with respect to $x$ is $\frac{d}{d x} [\frac{3 x}{16}] + \frac{d}{d x} [\frac{8}{x^{3}}]$ .

$\frac{d}{d x} [\frac{3 x}{16}] + \frac{d}{d x} [\frac{8}{x^{3}}]$

Step 1.2.2

Evaluate $\frac{d}{d x} [\frac{3 x}{16}]$ .

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Step 1.2.2.1

Since $\frac{3}{16}$ is constant with respect to $x$ , the derivative of $\frac{3 x}{16}$ with respect to $x$ is $\frac{3}{16} \frac{d}{d x} [x]$ .

$\frac{3}{16} \frac{d}{d x} [x] + \frac{d}{d x} [\frac{8}{x^{3}}]$

Step 1.2.2.2

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

$\frac{3}{16} \cdot 1 + \frac{d}{d x} [\frac{8}{x^{3}}]$

Step 1.2.2.3

Multiply $\frac{3}{16}$ by $1$ .

$\frac{3}{16} + \frac{d}{d x} [\frac{8}{x^{3}}]$

Step 1.2.3

Evaluate $\frac{d}{d x} [\frac{8}{x^{3}}]$ .

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Step 1.2.3.1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8}{x^{3}}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$\frac{3}{16} + 8 \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.2.3.2

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$\frac{3}{16} + 8 \frac{d}{d x} [{(x^{3})}^{- 1}]$

Step 1.2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{3}$ .

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Step 1.2.3.3.1

To apply the Chain Rule, set $u$ as $x^{3}$ .

$\frac{3}{16} + 8 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{3}])$

Step 1.2.3.3.2

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

$\frac{3}{16} + 8 (- u^{- 2} \frac{d}{d x} [x^{3}])$

Step 1.2.3.3.3

Replace all occurrences of $u$ with $x^{3}$ .

$\frac{3}{16} + 8 (- {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}])$

Step 1.2.3.4

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

$\frac{3}{16} + 8 (- {(x^{3})}^{- 2} (3 x^{2}))$

Step 1.2.3.5

Multiply the exponents in ${(x^{3})}^{- 2}$ .

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Step 1.2.3.5.1

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

$\frac{3}{16} + 8 (- x^{3 \cdot - 2} (3 x^{2}))$

Step 1.2.3.5.2

Multiply $3$ by $- 2$ .

$\frac{3}{16} + 8 (- x^{- 6} (3 x^{2}))$

Step 1.2.3.6

Multiply $3$ by $- 1$ .

$\frac{3}{16} + 8 (- 3 x^{- 6} x^{2})$

Step 1.2.3.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 1.2.3.7.1

Move $x^{2}$ .

$\frac{3}{16} + 8 (- 3 (x^{2} x^{- 6}))$

Step 1.2.3.7.2

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

$\frac{3}{16} + 8 (- 3 x^{2 - 6})$

Step 1.2.3.7.3

Subtract $6$ from $2$ .

$\frac{3}{16} + 8 (- 3 x^{- 4})$

Step 1.2.3.8

Multiply $- 3$ by $8$ .

$\frac{3}{16} - 24 x^{- 4}$

Step 1.2.4

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

$\frac{3}{16} - 24 \frac{1}{x^{4}}$

Step 1.2.5

Simplify.

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Step 1.2.5.1

Combine terms.

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Step 1.2.5.1.1

Combine $- 24$ and $\frac{1}{x^{4}}$ .

$\frac{3}{16} + \frac{- 24}{x^{4}}$

Step 1.2.5.1.2

Move the negative in front of the fraction.

$\frac{3}{16} - \frac{24}{x^{4}}$

Step 1.2.5.2

Reorder terms.

$f'' (x) = - \frac{24}{x^{4}} + \frac{3}{16}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{24}{x^{4}} + \frac{3}{16}$ .

$- \frac{24}{x^{4}} + \frac{3}{16}$

Step 2

Set the second derivative equal to $0$ then solve the equation $- \frac{24}{x^{4}} + \frac{3}{16} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$- \frac{24}{x^{4}} + \frac{3}{16} = 0$

Step 2.2

Subtract $\frac{3}{16}$ from both sides of the equation.

$- \frac{24}{x^{4}} = - \frac{3}{16}$

Step 2.3

Find the LCD of the terms in the equation.

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Step 2.3.1

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

$x^{4}, 16$

Step 2.3.2

Since $x^{4}, 16$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 16$ then find LCM for the variable part $x^{4}$ .

Step 2.3.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.3.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.3.5

The prime factors for $16$ are $2 \cdot 2 \cdot 2 \cdot 2$ .

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Step 2.3.5.1

$16$ has factors of $2$ and $8$ .

$2 \cdot 8$

Step 2.3.5.2

$8$ has factors of $2$ and $4$ .

$2 \cdot 2 \cdot 4$

Step 2.3.5.3

$4$ has factors of $2$ and $2$ .

$2 \cdot 2 \cdot 2 \cdot 2$

Step 2.3.6

Multiply $2 \cdot 2 \cdot 2 \cdot 2$ .

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Step 2.3.6.1

Multiply $2$ by $2$ .

$4 \cdot 2 \cdot 2$

Step 2.3.6.2

Multiply $4$ by $2$ .

$8 \cdot 2$

Step 2.3.6.3

Multiply $8$ by $2$ .

$16$

Step 2.3.7

The factors for $x^{4}$ are $x \cdot x \cdot x \cdot x$ , which is $x$ multiplied by each other $4$ times.

$x^{4} = x \cdot x \cdot x \cdot x$

$x$ occurs $4$ times.

Step 2.3.8

The LCM of $x^{4}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x \cdot x \cdot x$

Step 2.3.9

Simplify $x \cdot x \cdot x \cdot x$ .

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Step 2.3.9.1

Multiply $x$ by $x$ .

$x^{2} \cdot x \cdot x$

Step 2.3.9.2

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

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Step 2.3.9.2.1

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

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Step 2.3.9.2.1.1

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

$x^{2} \cdot x^{1} \cdot x$

Step 2.3.9.2.1.2

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

$x^{2 + 1} \cdot x$

Step 2.3.9.2.2

Add $2$ and $1$ .

$x^{3} \cdot x$

Step 2.3.9.3

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 2.3.9.3.1

Multiply $x^{3}$ by $x$ .

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Step 2.3.9.3.1.1

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

$x^{3} \cdot x^{1}$

Step 2.3.9.3.1.2

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

$x^{3 + 1}$

Step 2.3.9.3.2

Add $3$ and $1$ .

$x^{4}$

Step 2.3.10

The LCM for $x^{4}, 16$ is the numeric part $16$ multiplied by the variable part.

$16 x^{4}$

Step 2.4

Multiply each term in $- \frac{24}{x^{4}} = - \frac{3}{16}$ by $16 x^{4}$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $- \frac{24}{x^{4}} = - \frac{3}{16}$ by $16 x^{4}$ .

$- \frac{24}{x^{4}} (16 x^{4}) = - \frac{3}{16} (16 x^{4})$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $x^{4}$ .

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Step 2.4.2.1.1

Move the leading negative in $- \frac{24}{x^{4}}$ into the numerator.

$\frac{- 24}{x^{4}} (16 x^{4}) = - \frac{3}{16} (16 x^{4})$

Step 2.4.2.1.2

Factor $x^{4}$ out of $16 x^{4}$ .

$\frac{- 24}{x^{4}} (x^{4} \cdot 16) = - \frac{3}{16} (16 x^{4})$

Step 2.4.2.1.3

Cancel the common factor.

$\frac{- 24}{x^{4}} (x^{4} \cdot 16) = - \frac{3}{16} (16 x^{4})$

Step 2.4.2.1.4

Rewrite the expression.

$- 24 \cdot 16 = - \frac{3}{16} (16 x^{4})$

Step 2.4.2.2

Multiply $- 24$ by $16$ .

$- 384 = - \frac{3}{16} (16 x^{4})$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Cancel the common factor of $16$ .

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Step 2.4.3.1.1

Move the leading negative in $- \frac{3}{16}$ into the numerator.

$- 384 = \frac{- 3}{16} (16 x^{4})$

Step 2.4.3.1.2

Factor $16$ out of $16 x^{4}$ .

$- 384 = \frac{- 3}{16} (16 (x^{4}))$

Step 2.4.3.1.3

Cancel the common factor.

$- 384 = \frac{- 3}{16} (16 x^{4})$

Step 2.4.3.1.4

Rewrite the expression.

$- 384 = - 3 x^{4}$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $- 3 x^{4} = - 384$ .

$- 3 x^{4} = - 384$

Step 2.5.2

Divide each term in $- 3 x^{4} = - 384$ by $- 3$ and simplify.

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Step 2.5.2.1

Divide each term in $- 3 x^{4} = - 384$ by $- 3$ .

$\frac{- 3 x^{4}}{- 3} = \frac{- 384}{- 3}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $- 3$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{- 3 x^{4}}{- 3} = \frac{- 384}{- 3}$

Step 2.5.2.2.1.2

Divide $x^{4}$ by $1$ .

$x^{4} = \frac{- 384}{- 3}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Divide $- 384$ by $- 3$ .

$x^{4} = 128$

Step 2.5.3

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

$x = \pm \sqrt[4]{128}$

Step 2.5.4

Simplify $\pm \sqrt[4]{128}$ .

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Step 2.5.4.1

Rewrite $128$ as $2^{4} \cdot 8$ .

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Step 2.5.4.1.1

Factor $16$ out of $128$ .

$x = \pm \sqrt[4]{16 (8)}$

Step 2.5.4.1.2

Rewrite $16$ as $2^{4}$ .

$x = \pm \sqrt[4]{2^{4} \cdot 8}$

Step 2.5.4.2

Pull terms out from under the radical.

$x = \pm 2 \sqrt[4]{8}$

Step 2.5.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.5.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2 \sqrt[4]{8}$

Step 2.5.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2 \sqrt[4]{8}$

Step 2.5.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2 \sqrt[4]{8}, - 2 \sqrt[4]{8}$

Step 3

Find the points where the second derivative is $0$ .

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Step 3.1

Substitute $2 \sqrt[4]{8}$ in $f (x) = \frac{3}{32} x^{2} - 4 x^{- 2}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $2 \sqrt[4]{8}$ in the expression.

$f (2 \sqrt[4]{8}) = \frac{3}{32} \cdot {(2 \sqrt[4]{8})}^{2} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Apply the product rule to $2 \sqrt[4]{8}$ .

$f (2 \sqrt[4]{8}) = \frac{3}{32} \cdot (2^{2} {\sqrt[4]{8}}^{2}) - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.2

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

$f (2 \sqrt[4]{8}) = \frac{3}{32} \cdot (4 {\sqrt[4]{8}}^{2}) - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.3

Cancel the common factor of $4$ .

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Step 3.1.2.1.3.1

Factor $4$ out of $32$ .

$f (2 \sqrt[4]{8}) = \frac{3}{4 (8)} \cdot (4 {\sqrt[4]{8}}^{2}) - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.3.2

Factor $4$ out of $4 {\sqrt[4]{8}}^{2}$ .

$f (2 \sqrt[4]{8}) = \frac{3}{4 (8)} \cdot (4 ({\sqrt[4]{8}}^{2})) - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.3.3

Cancel the common factor.

$f (2 \sqrt[4]{8}) = \frac{3}{4 \cdot 8} \cdot (4 {\sqrt[4]{8}}^{2}) - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.3.4

Rewrite the expression.

$f (2 \sqrt[4]{8}) = \frac{3}{8} \cdot {\sqrt[4]{8}}^{2} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.4

Combine $\frac{3}{8}$ and ${\sqrt[4]{8}}^{2}$ .

$f (2 \sqrt[4]{8}) = \frac{3 {\sqrt[4]{8}}^{2}}{8} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.5

Simplify the numerator.

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Step 3.1.2.1.5.1

Rewrite ${\sqrt[4]{8}}^{2}$ as $\sqrt{8}$ .

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Step 3.1.2.1.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{8}$ as $8^{\frac{1}{4}}$ .

$f (2 \sqrt[4]{8}) = \frac{3 {(8^{\frac{1}{4}})}^{2}}{8} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.5.1.2

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

$f (2 \sqrt[4]{8}) = \frac{3 \cdot 8^{\frac{1}{4} \cdot 2}}{8} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.5.1.3

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

$f (2 \sqrt[4]{8}) = \frac{3 \cdot 8^{\frac{2}{4}}}{8} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.5.1.4

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

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Step 3.1.2.1.5.1.4.1

Factor $2$ out of $2$ .

$f (2 \sqrt[4]{8}) = \frac{3 \cdot 8^{\frac{2 (1)}{4}}}{8} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.5.1.4.2

Cancel the common factors.

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Step 3.1.2.1.5.1.4.2.1

Factor $2$ out of $4$ .

$f (2 \sqrt[4]{8}) = \frac{3 \cdot 8^{\frac{2 \cdot 1}{2 \cdot 2}}}{8} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.5.1.4.2.2

Cancel the common factor.

$f (2 \sqrt[4]{8}) = \frac{3 \cdot 8^{\frac{2 \cdot 1}{2 \cdot 2}}}{8} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.5.1.4.2.3

Rewrite the expression.

$f (2 \sqrt[4]{8}) = \frac{3 \cdot 8^{\frac{1}{2}}}{8} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.5.1.5

Rewrite $8^{\frac{1}{2}}$ as $\sqrt{8}$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{8}}{8} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.5.2

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.1.2.1.5.2.1

Factor $4$ out of $8$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{4 (2)}}{8} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.5.2.2

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

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2^{2} \cdot 2}}{8} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.5.3

Pull terms out from under the radical.

$f (2 \sqrt[4]{8}) = \frac{3 \cdot (2 \sqrt{2})}{8} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.5.4

Multiply $3$ by $2$ .

$f (2 \sqrt[4]{8}) = \frac{6 \sqrt{2}}{8} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.6

Cancel the common factor of $6$ and $8$ .

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Step 3.1.2.1.6.1

Factor $2$ out of $6 \sqrt{2}$ .

$f (2 \sqrt[4]{8}) = \frac{2 (3 \sqrt{2})}{8} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.6.2

Cancel the common factors.

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Step 3.1.2.1.6.2.1

Factor $2$ out of $8$ .

$f (2 \sqrt[4]{8}) = \frac{2 (3 \sqrt{2})}{2 (4)} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.6.2.2

Cancel the common factor.

$f (2 \sqrt[4]{8}) = \frac{2 (3 \sqrt{2})}{2 \cdot 4} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.6.2.3

Rewrite the expression.

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 {(2 \sqrt[4]{8})}^{- 2}$

Step 3.1.2.1.7

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

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{{(2 \sqrt[4]{8})}^{2}}$

Step 3.1.2.1.8

Simplify the denominator.

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Step 3.1.2.1.8.1

Apply the product rule to $2 \sqrt[4]{8}$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{2^{2} {\sqrt[4]{8}}^{2}}$

Step 3.1.2.1.8.2

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

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 {\sqrt[4]{8}}^{2}}$

Step 3.1.2.1.8.3

Rewrite ${\sqrt[4]{8}}^{2}$ as $\sqrt{8}$ .

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Step 3.1.2.1.8.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{8}$ as $8^{\frac{1}{4}}$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 {(8^{\frac{1}{4}})}^{2}}$

Step 3.1.2.1.8.3.2

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

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \cdot 8^{\frac{1}{4} \cdot 2}}$

Step 3.1.2.1.8.3.3

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

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \cdot 8^{\frac{2}{4}}}$

Step 3.1.2.1.8.3.4

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

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Step 3.1.2.1.8.3.4.1

Factor $2$ out of $2$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \cdot 8^{\frac{2 (1)}{4}}}$

Step 3.1.2.1.8.3.4.2

Cancel the common factors.

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Step 3.1.2.1.8.3.4.2.1

Factor $2$ out of $4$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \cdot 8^{\frac{2 \cdot 1}{2 \cdot 2}}}$

Step 3.1.2.1.8.3.4.2.2

Cancel the common factor.

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \cdot 8^{\frac{2 \cdot 1}{2 \cdot 2}}}$

Step 3.1.2.1.8.3.4.2.3

Rewrite the expression.

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \cdot 8^{\frac{1}{2}}}$

Step 3.1.2.1.8.3.5

Rewrite $8^{\frac{1}{2}}$ as $\sqrt{8}$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \sqrt{8}}$

Step 3.1.2.1.8.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.1.2.1.8.4.1

Factor $4$ out of $8$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \sqrt{4 (2)}}$

Step 3.1.2.1.8.4.2

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

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \sqrt{2^{2} \cdot 2}}$

Step 3.1.2.1.8.5

Pull terms out from under the radical.

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \cdot (2 \sqrt{2})}$

Step 3.1.2.1.8.6

Multiply $4$ by $2$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{8 \sqrt{2}}$

Step 3.1.2.1.9

Cancel the common factor of $4$ .

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Step 3.1.2.1.9.1

Factor $4$ out of $- 4$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} + 4 (- 1) (\frac{1}{8 \sqrt{2}})$

Step 3.1.2.1.9.2

Factor $4$ out of $8 \sqrt{2}$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} + 4 (- 1) (\frac{1}{4 (2 \sqrt{2})})$

Step 3.1.2.1.9.3

Cancel the common factor.

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} + 4 \cdot (- 1 \frac{1}{4 (2 \sqrt{2})})$

Step 3.1.2.1.9.4

Rewrite the expression.

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{1}{2 \sqrt{2}}$

Step 3.1.2.1.10

Multiply $\frac{1}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 (\frac{1}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 3.1.2.1.11

Combine and simplify the denominator.

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Step 3.1.2.1.11.1

Multiply $\frac{1}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 3.1.2.1.11.2

Move $\sqrt{2}$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 3.1.2.1.11.3

Raise $\sqrt{2}$ to the power of $1$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 3.1.2.1.11.4

Raise $\sqrt{2}$ to the power of $1$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 3.1.2.1.11.5

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

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 3.1.2.1.11.6

Add $1$ and $1$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 3.1.2.1.11.7

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

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Step 3.1.2.1.11.7.1

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

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Step 3.1.2.1.11.7.2

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

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 3.1.2.1.11.7.3

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

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 3.1.2.1.11.7.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.11.7.4.1

Cancel the common factor.

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 3.1.2.1.11.7.4.2

Rewrite the expression.

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 \cdot 2}$

Step 3.1.2.1.11.7.5

Evaluate the exponent.

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 \cdot 2}$

Step 3.1.2.1.12

Multiply $2$ by $2$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{4}$

Step 3.1.2.1.13

Rewrite $- 1 \frac{\sqrt{2}}{4}$ as $- \frac{\sqrt{2}}{4}$ .

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - \frac{\sqrt{2}}{4}$

Step 3.1.2.2

Simplify terms.

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Step 3.1.2.2.1

Combine the numerators over the common denominator.

$f (2 \sqrt[4]{8}) = \frac{3 \sqrt{2} - \sqrt{2}}{4}$

Step 3.1.2.2.2

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

$f (2 \sqrt[4]{8}) = \frac{2 \sqrt{2}}{4}$

Step 3.1.2.2.3

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

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Step 3.1.2.2.3.1

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

$f (2 \sqrt[4]{8}) = \frac{2 (\sqrt{2})}{4}$

Step 3.1.2.2.3.2

Cancel the common factors.

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Step 3.1.2.2.3.2.1

Factor $2$ out of $4$ .

$f (2 \sqrt[4]{8}) = \frac{2 \sqrt{2}}{2 \cdot 2}$

Step 3.1.2.2.3.2.2

Cancel the common factor.

$f (2 \sqrt[4]{8}) = \frac{2 \sqrt{2}}{2 \cdot 2}$

Step 3.1.2.2.3.2.3

Rewrite the expression.

$f (2 \sqrt[4]{8}) = \frac{\sqrt{2}}{2}$

Step 3.1.2.3

The final answer is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2}$

Step 3.2

The point found by substituting $2 \sqrt[4]{8}$ in $f (x) = \frac{3}{32} x^{2} - 4 x^{- 2}$ is $(2 \sqrt[4]{8}, \frac{\sqrt{2}}{2})$ . This point can be an inflection point.

$(2 \sqrt[4]{8}, \frac{\sqrt{2}}{2})$

Step 3.3

Substitute $- 2 \sqrt[4]{8}$ in $f (x) = \frac{3}{32} x^{2} - 4 x^{- 2}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $- 2 \sqrt[4]{8}$ in the expression.

$f (- 2 \sqrt[4]{8}) = \frac{3}{32} \cdot {(- 2 \sqrt[4]{8})}^{2} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Apply the product rule to $- 2 \sqrt[4]{8}$ .

$f (- 2 \sqrt[4]{8}) = \frac{3}{32} \cdot ({(- 2)}^{2} {\sqrt[4]{8}}^{2}) - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.2

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

$f (- 2 \sqrt[4]{8}) = \frac{3}{32} \cdot (4 {\sqrt[4]{8}}^{2}) - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.3

Cancel the common factor of $4$ .

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Step 3.3.2.1.3.1

Factor $4$ out of $32$ .

$f (- 2 \sqrt[4]{8}) = \frac{3}{4 (8)} \cdot (4 {\sqrt[4]{8}}^{2}) - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.3.2

Factor $4$ out of $4 {\sqrt[4]{8}}^{2}$ .

$f (- 2 \sqrt[4]{8}) = \frac{3}{4 (8)} \cdot (4 ({\sqrt[4]{8}}^{2})) - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.3.3

Cancel the common factor.

$f (- 2 \sqrt[4]{8}) = \frac{3}{4 \cdot 8} \cdot (4 {\sqrt[4]{8}}^{2}) - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.3.4

Rewrite the expression.

$f (- 2 \sqrt[4]{8}) = \frac{3}{8} \cdot {\sqrt[4]{8}}^{2} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.4

Combine $\frac{3}{8}$ and ${\sqrt[4]{8}}^{2}$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 {\sqrt[4]{8}}^{2}}{8} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.5

Simplify the numerator.

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Step 3.3.2.1.5.1

Rewrite ${\sqrt[4]{8}}^{2}$ as $\sqrt{8}$ .

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Step 3.3.2.1.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{8}$ as $8^{\frac{1}{4}}$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 {(8^{\frac{1}{4}})}^{2}}{8} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.5.1.2

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

$f (- 2 \sqrt[4]{8}) = \frac{3 \cdot 8^{\frac{1}{4} \cdot 2}}{8} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.5.1.3

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

$f (- 2 \sqrt[4]{8}) = \frac{3 \cdot 8^{\frac{2}{4}}}{8} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.5.1.4

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

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Step 3.3.2.1.5.1.4.1

Factor $2$ out of $2$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \cdot 8^{\frac{2 (1)}{4}}}{8} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.5.1.4.2

Cancel the common factors.

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Step 3.3.2.1.5.1.4.2.1

Factor $2$ out of $4$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \cdot 8^{\frac{2 \cdot 1}{2 \cdot 2}}}{8} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.5.1.4.2.2

Cancel the common factor.

$f (- 2 \sqrt[4]{8}) = \frac{3 \cdot 8^{\frac{2 \cdot 1}{2 \cdot 2}}}{8} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.5.1.4.2.3

Rewrite the expression.

$f (- 2 \sqrt[4]{8}) = \frac{3 \cdot 8^{\frac{1}{2}}}{8} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.5.1.5

Rewrite $8^{\frac{1}{2}}$ as $\sqrt{8}$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{8}}{8} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.5.2

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.3.2.1.5.2.1

Factor $4$ out of $8$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{4 (2)}}{8} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.5.2.2

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

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2^{2} \cdot 2}}{8} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.5.3

Pull terms out from under the radical.

$f (- 2 \sqrt[4]{8}) = \frac{3 \cdot (2 \sqrt{2})}{8} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.5.4

Multiply $3$ by $2$ .

$f (- 2 \sqrt[4]{8}) = \frac{6 \sqrt{2}}{8} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.6

Cancel the common factor of $6$ and $8$ .

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Step 3.3.2.1.6.1

Factor $2$ out of $6 \sqrt{2}$ .

$f (- 2 \sqrt[4]{8}) = \frac{2 (3 \sqrt{2})}{8} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.6.2

Cancel the common factors.

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Step 3.3.2.1.6.2.1

Factor $2$ out of $8$ .

$f (- 2 \sqrt[4]{8}) = \frac{2 (3 \sqrt{2})}{2 (4)} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.6.2.2

Cancel the common factor.

$f (- 2 \sqrt[4]{8}) = \frac{2 (3 \sqrt{2})}{2 \cdot 4} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.6.2.3

Rewrite the expression.

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 {(- 2 \sqrt[4]{8})}^{- 2}$

Step 3.3.2.1.7

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

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{{(- 2 \sqrt[4]{8})}^{2}}$

Step 3.3.2.1.8

Simplify the denominator.

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Step 3.3.2.1.8.1

Apply the product rule to $- 2 \sqrt[4]{8}$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{{(- 2)}^{2} {\sqrt[4]{8}}^{2}}$

Step 3.3.2.1.8.2

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

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 {\sqrt[4]{8}}^{2}}$

Step 3.3.2.1.8.3

Rewrite ${\sqrt[4]{8}}^{2}$ as $\sqrt{8}$ .

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Step 3.3.2.1.8.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{8}$ as $8^{\frac{1}{4}}$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 {(8^{\frac{1}{4}})}^{2}}$

Step 3.3.2.1.8.3.2

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

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \cdot 8^{\frac{1}{4} \cdot 2}}$

Step 3.3.2.1.8.3.3

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

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \cdot 8^{\frac{2}{4}}}$

Step 3.3.2.1.8.3.4

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

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Step 3.3.2.1.8.3.4.1

Factor $2$ out of $2$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \cdot 8^{\frac{2 (1)}{4}}}$

Step 3.3.2.1.8.3.4.2

Cancel the common factors.

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Step 3.3.2.1.8.3.4.2.1

Factor $2$ out of $4$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \cdot 8^{\frac{2 \cdot 1}{2 \cdot 2}}}$

Step 3.3.2.1.8.3.4.2.2

Cancel the common factor.

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \cdot 8^{\frac{2 \cdot 1}{2 \cdot 2}}}$

Step 3.3.2.1.8.3.4.2.3

Rewrite the expression.

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \cdot 8^{\frac{1}{2}}}$

Step 3.3.2.1.8.3.5

Rewrite $8^{\frac{1}{2}}$ as $\sqrt{8}$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \sqrt{8}}$

Step 3.3.2.1.8.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.3.2.1.8.4.1

Factor $4$ out of $8$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \sqrt{4 (2)}}$

Step 3.3.2.1.8.4.2

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

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \sqrt{2^{2} \cdot 2}}$

Step 3.3.2.1.8.5

Pull terms out from under the radical.

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{4 \cdot (2 \sqrt{2})}$

Step 3.3.2.1.8.6

Multiply $4$ by $2$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 4 \frac{1}{8 \sqrt{2}}$

Step 3.3.2.1.9

Cancel the common factor of $4$ .

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Step 3.3.2.1.9.1

Factor $4$ out of $- 4$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} + 4 (- 1) (\frac{1}{8 \sqrt{2}})$

Step 3.3.2.1.9.2

Factor $4$ out of $8 \sqrt{2}$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} + 4 (- 1) (\frac{1}{4 (2 \sqrt{2})})$

Step 3.3.2.1.9.3

Cancel the common factor.

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} + 4 \cdot (- 1 \frac{1}{4 (2 \sqrt{2})})$

Step 3.3.2.1.9.4

Rewrite the expression.

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{1}{2 \sqrt{2}}$

Step 3.3.2.1.10

Multiply $\frac{1}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 (\frac{1}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 3.3.2.1.11

Combine and simplify the denominator.

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Step 3.3.2.1.11.1

Multiply $\frac{1}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 3.3.2.1.11.2

Move $\sqrt{2}$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 3.3.2.1.11.3

Raise $\sqrt{2}$ to the power of $1$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 3.3.2.1.11.4

Raise $\sqrt{2}$ to the power of $1$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 3.3.2.1.11.5

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

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 3.3.2.1.11.6

Add $1$ and $1$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 3.3.2.1.11.7

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

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Step 3.3.2.1.11.7.1

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

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Step 3.3.2.1.11.7.2

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

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 3.3.2.1.11.7.3

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

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 3.3.2.1.11.7.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.11.7.4.1

Cancel the common factor.

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 3.3.2.1.11.7.4.2

Rewrite the expression.

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 \cdot 2}$

Step 3.3.2.1.11.7.5

Evaluate the exponent.

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{2 \cdot 2}$

Step 3.3.2.1.12

Multiply $2$ by $2$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - 1 \frac{\sqrt{2}}{4}$

Step 3.3.2.1.13

Rewrite $- 1 \frac{\sqrt{2}}{4}$ as $- \frac{\sqrt{2}}{4}$ .

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2}}{4} - \frac{\sqrt{2}}{4}$

Step 3.3.2.2

Simplify terms.

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Step 3.3.2.2.1

Combine the numerators over the common denominator.

$f (- 2 \sqrt[4]{8}) = \frac{3 \sqrt{2} - \sqrt{2}}{4}$

Step 3.3.2.2.2

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

$f (- 2 \sqrt[4]{8}) = \frac{2 \sqrt{2}}{4}$

Step 3.3.2.2.3

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

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Step 3.3.2.2.3.1

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

$f (- 2 \sqrt[4]{8}) = \frac{2 (\sqrt{2})}{4}$

Step 3.3.2.2.3.2

Cancel the common factors.

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Step 3.3.2.2.3.2.1

Factor $2$ out of $4$ .

$f (- 2 \sqrt[4]{8}) = \frac{2 \sqrt{2}}{2 \cdot 2}$

Step 3.3.2.2.3.2.2

Cancel the common factor.

$f (- 2 \sqrt[4]{8}) = \frac{2 \sqrt{2}}{2 \cdot 2}$

Step 3.3.2.2.3.2.3

Rewrite the expression.

$f (- 2 \sqrt[4]{8}) = \frac{\sqrt{2}}{2}$

Step 3.3.2.3

The final answer is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2}$

Step 3.4

The point found by substituting $- 2 \sqrt[4]{8}$ in $f (x) = \frac{3}{32} x^{2} - 4 x^{- 2}$ is $(- 2 \sqrt[4]{8}, \frac{\sqrt{2}}{2})$ . This point can be an inflection point.

$(- 2 \sqrt[4]{8}, \frac{\sqrt{2}}{2})$

Step 3.5

Determine the points that could be inflection points.

$(2 \sqrt[4]{8}, \frac{\sqrt{2}}{2}), (- 2 \sqrt[4]{8}, \frac{\sqrt{2}}{2})$

Step 4

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, - 2 \sqrt[4]{8}) \cup (- 2 \sqrt[4]{8}, 2 \sqrt[4]{8}) \cup (2 \sqrt[4]{8}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 2 \sqrt[4]{8})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 3.46358566$ in the expression.

$f'' (- 3.46358566) = - \frac{24}{{(- 3.46358566)}^{4}} + \frac{3}{16}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $- 3.46358566$ to the power of $4$ .

$f'' (- 3.46358566) = - \frac{24}{143.91422792} + \frac{3}{16}$

Step 5.2.1.2

Divide $24$ by $143.91422792$ .

$f'' (- 3.46358566) = - 1 \cdot 0.16676599 + \frac{3}{16}$

Step 5.2.1.3

Multiply $- 1$ by $0.16676599$ .

$f'' (- 3.46358566) = - 0.16676599 + \frac{3}{16}$

Step 5.2.2

To write $- 0.16676599$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$f'' (- 3.46358566) = - 0.16676599 \cdot \frac{16}{16} + \frac{3}{16}$

Step 5.2.3

Combine fractions.

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Step 5.2.3.1

Combine $- 0.16676599$ and $\frac{16}{16}$ .

$f'' (- 3.46358566) = \frac{- 0.16676599 \cdot 16}{16} + \frac{3}{16}$

Step 5.2.3.2

Combine the numerators over the common denominator.

$f'' (- 3.46358566) = \frac{- 0.16676599 \cdot 16 + 3}{16}$

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

Multiply $- 0.16676599$ by $16$ .

$f'' (- 3.46358566) = \frac{- 2.66825598 + 3}{16}$

Step 5.2.4.2

Add $- 2.66825598$ and $3$ .

$f'' (- 3.46358566) = \frac{0.33174401}{16}$

Step 5.2.5

Divide $0.33174401$ by $16$ .

$f'' (- 3.46358566) = 0.020734$

Step 5.2.6

The final answer is $0.020734$ .

$0.020734$

Step 5.3

At $- 3.46358566$ , the second derivative is $0.020734$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - 2 \sqrt[4]{8})$ .

Increasing on $(- \infty, - 2 \sqrt[4]{8})$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(- 2 \sqrt[4]{8}, 2 \sqrt[4]{8})$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $0$ in the expression.

$f'' (0) = - \frac{24}{{(0)}^{4}} + \frac{3}{16}$

Step 6.2

Raising $0$ to any positive power yields $0$ .

$f'' (0) = - \frac{24}{0} + \frac{3}{16}$

Step 6.3

The expression contains a division by $0$ . The expression is undefined.

$f'' (0) = Undefined$

Step 6.4

At $0$ , the second derivative is $N a N$ . Since this is negative, the second derivative is decreasing on the interval $(- 2 \sqrt[4]{8}, 2 \sqrt[4]{8})$

Decreasing on $(- 2 \sqrt[4]{8}, 2 \sqrt[4]{8})$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(2 \sqrt[4]{8}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $3.46358566$ in the expression.

$f'' (3.46358566) = - \frac{24}{{(3.46358566)}^{4}} + \frac{3}{16}$

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

Raise $3.46358566$ to the power of $4$ .

$f'' (3.46358566) = - \frac{24}{143.91422792} + \frac{3}{16}$

Step 7.2.1.2

Divide $24$ by $143.91422792$ .

$f'' (3.46358566) = - 1 \cdot 0.16676599 + \frac{3}{16}$

Step 7.2.1.3

Multiply $- 1$ by $0.16676599$ .

$f'' (3.46358566) = - 0.16676599 + \frac{3}{16}$

Step 7.2.2

To write $- 0.16676599$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$f'' (3.46358566) = - 0.16676599 \cdot \frac{16}{16} + \frac{3}{16}$

Step 7.2.3

Combine fractions.

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Step 7.2.3.1

Combine $- 0.16676599$ and $\frac{16}{16}$ .

$f'' (3.46358566) = \frac{- 0.16676599 \cdot 16}{16} + \frac{3}{16}$

Step 7.2.3.2

Combine the numerators over the common denominator.

$f'' (3.46358566) = \frac{- 0.16676599 \cdot 16 + 3}{16}$

Step 7.2.4

Simplify the numerator.

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Step 7.2.4.1

Multiply $- 0.16676599$ by $16$ .

$f'' (3.46358566) = \frac{- 2.66825598 + 3}{16}$

Step 7.2.4.2

Add $- 2.66825598$ and $3$ .

$f'' (3.46358566) = \frac{0.33174401}{16}$

Step 7.2.5

Divide $0.33174401$ by $16$ .

$f'' (3.46358566) = 0.020734$

Step 7.2.6

The final answer is $0.020734$ .

$0.020734$

Step 7.3

At $3.46358566$ , the second derivative is $0.020734$ . Since this is positive, the second derivative is increasing on the interval $(2 \sqrt[4]{8}, \infty)$ .

Increasing on $(2 \sqrt[4]{8}, \infty)$ since $f'' (x) > 0$

Step 8

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are $(- 2 \sqrt[4]{8}, \frac{\sqrt{2}}{2}), (2 \sqrt[4]{8}, \frac{\sqrt{2}}{2})$ .

$(- 2 \sqrt[4]{8}, \frac{\sqrt{2}}{2}), (2 \sqrt[4]{8}, \frac{\sqrt{2}}{2})$

Step 9