Calculus Examples

$f (x) = \frac{2}{x^{4} - 16}$ on interval $- 1$ , $1.5$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1.1.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{x^{4} - 16}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{x^{4} - 16}]$ .

$2 \frac{d}{d x} [\frac{1}{x^{4} - 16}]$

Step 1.1.1.1.2

Rewrite $\frac{1}{x^{4} - 16}$ as ${(x^{4} - 16)}^{- 1}$ .

$2 \frac{d}{d x} [{(x^{4} - 16)}^{- 1}]$

Step 1.1.1.2

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 $g (x) = x^{4} - 16$ .

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Step 1.1.1.2.1

To apply the Chain Rule, set $u$ as $x^{4} - 16$ .

$2 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{4} - 16])$

Step 1.1.1.2.2

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

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

Step 1.1.1.2.3

Replace all occurrences of $u$ with $x^{4} - 16$ .

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

Step 1.1.1.3

Differentiate.

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Step 1.1.1.3.1

Multiply $- 1$ by $2$ .

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

Step 1.1.1.3.2

By the Sum Rule, the derivative of $x^{4} - 16$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 16]$ .

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

Step 1.1.1.3.3

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

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

Step 1.1.1.3.4

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

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

Step 1.1.1.3.5

Simplify the expression.

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Step 1.1.1.3.5.1

Add $4 x^{3}$ and $0$ .

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

Step 1.1.1.3.5.2

Multiply $4$ by $- 2$ .

$- 8 {(x^{4} - 16)}^{- 2} x^{3}$

Step 1.1.1.4

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

$- 8 \frac{1}{{(x^{4} - 16)}^{2}} x^{3}$

Step 1.1.1.5

Combine terms.

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Step 1.1.1.5.1

Combine $- 8$ and $\frac{1}{{(x^{4} - 16)}^{2}}$ .

$\frac{- 8}{{(x^{4} - 16)}^{2}} x^{3}$

Step 1.1.1.5.2

Move the negative in front of the fraction.

$- \frac{8}{{(x^{4} - 16)}^{2}} x^{3}$

Step 1.1.1.5.3

Combine $x^{3}$ and $\frac{8}{{(x^{4} - 16)}^{2}}$ .

$- \frac{x^{3} \cdot 8}{{(x^{4} - 16)}^{2}}$

Step 1.1.1.5.4

Move $8$ to the left of $x^{3}$ .

$f' (x) = - \frac{8 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{8 x^{3}}{{(x^{4} - 16)}^{2}}$ .

$- \frac{8 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $- \frac{8 x^{3}}{{(x^{4} - 16)}^{2}} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$- \frac{8 x^{3}}{{(x^{4} - 16)}^{2}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$8 x^{3} = 0$

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

Divide each term in $8 x^{3} = 0$ by $8$ and simplify.

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Step 1.2.3.1.1

Divide each term in $8 x^{3} = 0$ by $8$ .

$\frac{8 x^{3}}{8} = \frac{0}{8}$

Step 1.2.3.1.2

Simplify the left side.

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Step 1.2.3.1.2.1

Cancel the common factor of $8$ .

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Step 1.2.3.1.2.1.1

Cancel the common factor.

$\frac{8 x^{3}}{8} = \frac{0}{8}$

Step 1.2.3.1.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{0}{8}$

Step 1.2.3.1.3

Simplify the right side.

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Step 1.2.3.1.3.1

Divide $0$ by $8$ .

$x^{3} = 0$

Step 1.2.3.2

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

$x = \sqrt[3]{0}$

Step 1.2.3.3

Simplify $\sqrt[3]{0}$ .

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Step 1.2.3.3.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 1.2.3.3.2

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

$x = 0$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Set the denominator in $\frac{8 x^{3}}{{(x^{4} - 16)}^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 1.3.2

Solve for $x$ .

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Step 1.3.2.1

Factor the left side of the equation.

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Step 1.3.2.1.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

${({(x^{2})}^{2} - 16)}^{2} = 0$

Step 1.3.2.1.2

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

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

Step 1.3.2.1.3

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

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

Step 1.3.2.1.4

Simplify.

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Step 1.3.2.1.4.1

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

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

Step 1.3.2.1.4.2

Factor.

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Step 1.3.2.1.4.2.1

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

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

Step 1.3.2.1.4.2.2

Remove unnecessary parentheses.

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

Step 1.3.2.1.5

Apply the product rule to $(x^{2} + 4) (x + 2) (x - 2)$ .

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

Step 1.3.2.1.6

Apply the product rule to $(x^{2} + 4) (x + 2)$ .

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

Step 1.3.2.2

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

${(x^{2} + 4)}^{2} = 0$

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

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

Step 1.3.2.3

Set ${(x^{2} + 4)}^{2}$ equal to $0$ and solve for $x$ .

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Step 1.3.2.3.1

Set ${(x^{2} + 4)}^{2}$ equal to $0$ .

${(x^{2} + 4)}^{2} = 0$

Step 1.3.2.3.2

Solve ${(x^{2} + 4)}^{2} = 0$ for $x$ .

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Step 1.3.2.3.2.1

Set the $x^{2} + 4$ equal to $0$ .

$x^{2} + 4 = 0$

Step 1.3.2.3.2.2

Solve for $x$ .

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Step 1.3.2.3.2.2.1

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

Step 1.3.2.3.2.2.2

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

$x = \pm \sqrt{- 4}$

Step 1.3.2.3.2.2.3

Simplify $\pm \sqrt{- 4}$ .

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Step 1.3.2.3.2.2.3.1

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Step 1.3.2.3.2.2.3.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 1.3.2.3.2.2.3.3

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

$x = \pm i \cdot \sqrt{4}$

Step 1.3.2.3.2.2.3.4

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

$x = \pm i \cdot \sqrt{2^{2}}$

Step 1.3.2.3.2.2.3.5

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

$x = \pm i \cdot 2$

Step 1.3.2.3.2.2.3.6

Move $2$ to the left of $i$ .

$x = \pm 2 i$

Step 1.3.2.3.2.2.4

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

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Step 1.3.2.3.2.2.4.1

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

$x = 2 i$

Step 1.3.2.3.2.2.4.2

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

$x = - 2 i$

Step 1.3.2.3.2.2.4.3

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

$x = 2 i, - 2 i$

Step 1.3.2.4

Set ${(x + 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 1.3.2.4.1

Set ${(x + 2)}^{2}$ equal to $0$ .

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

Step 1.3.2.4.2

Solve ${(x + 2)}^{2} = 0$ for $x$ .

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Step 1.3.2.4.2.1

Set the $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 1.3.2.4.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.3.2.5

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

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Step 1.3.2.5.1

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

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

Step 1.3.2.5.2

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

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Step 1.3.2.5.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 1.3.2.5.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.3.2.6

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

$x = 2 i, - 2 i, - 2, 2$

Step 1.3.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 2, x = 2$

Step 1.4

Evaluate $\frac{2}{x^{4} - 16}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 0$ .

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Step 1.4.1.1

Substitute $0$ for $x$ .

$\frac{2}{{(0)}^{4} - 16}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify the denominator.

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Step 1.4.1.2.1.1

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

$\frac{2}{0 - 16}$

Step 1.4.1.2.1.2

Subtract $16$ from $0$ .

$\frac{2}{- 16}$

Step 1.4.1.2.2

Reduce the expression by cancelling the common factors.

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Step 1.4.1.2.2.1

Cancel the common factor of $2$ and $- 16$ .

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Step 1.4.1.2.2.1.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{- 16}$

Step 1.4.1.2.2.1.2

Cancel the common factors.

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Step 1.4.1.2.2.1.2.1

Factor $2$ out of $- 16$ .

$\frac{2 \cdot 1}{2 \cdot - 8}$

Step 1.4.1.2.2.1.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot - 8}$

Step 1.4.1.2.2.1.2.3

Rewrite the expression.

$\frac{1}{- 8}$

Step 1.4.1.2.2.2

Move the negative in front of the fraction.

$- \frac{1}{8}$

Step 1.4.2

Evaluate at $x = - 2$ .

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Step 1.4.2.1

Substitute $- 2$ for $x$ .

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

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

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

$\frac{2}{16 - 16}$

Step 1.4.2.2.2

Subtract $16$ from $16$ .

$\frac{2}{0}$

Step 1.4.2.2.3

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

Undefined

Step 1.4.3

Evaluate at $x = 2$ .

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Step 1.4.3.1

Substitute $2$ for $x$ .

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

Step 1.4.3.2

Simplify.

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Step 1.4.3.2.1

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

$\frac{2}{16 - 16}$

Step 1.4.3.2.2

Subtract $16$ from $16$ .

$\frac{2}{0}$

Step 1.4.3.2.3

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

Undefined

Step 1.4.4

List all of the points.

$(0, - \frac{1}{8})$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = - 1$ .

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Step 2.1.1

Substitute $- 1$ for $x$ .

$\frac{2}{{(- 1)}^{4} - 16}$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Simplify the denominator.

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Step 2.1.2.1.1

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

$\frac{2}{1 - 16}$

Step 2.1.2.1.2

Subtract $16$ from $1$ .

$\frac{2}{- 15}$

Step 2.1.2.2

Move the negative in front of the fraction.

$- \frac{2}{15}$

Step 2.2

Evaluate at $x = 1.5$ .

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Step 2.2.1

Substitute $1.5$ for $x$ .

$\frac{2}{{(1.5)}^{4} - 16}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Simplify the denominator.

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Step 2.2.2.1.1

Raise $1.5$ to the power of $4$ .

$\frac{2}{5.0625 - 16}$

Step 2.2.2.1.2

Subtract $16$ from $5.0625$ .

$\frac{2}{- 10.9375}$

Step 2.2.2.2

Divide $2$ by $- 10.9375$ .

$- 0.18285714$

Step 2.3

List all of the points.

$(- 1, - \frac{2}{15}), (1.5, - 0.18285714)$

Step 3

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(0, - \frac{1}{8})$

Absolute Minimum: $(1.5, - 0.18285714)$

Step 4