Calculus Examples

$f (x) = x^{4} - 4 x^{2}$ , $[- 3, 4]$

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.

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Step 1.1.1.1.1

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

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

Step 1.1.1.1.2

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

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

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

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

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

Step 1.1.1.2.3

Multiply $2$ by $- 4$ .

$f' (x) = 4 x^{3} - 8 x$

Step 1.1.2

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

$4 x^{3} - 8 x$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $4 x^{3} - 8 x = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$4 x^{3} - 8 x = 0$

Step 1.2.2

Factor $4 x$ out of $4 x^{3} - 8 x$ .

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Step 1.2.2.1

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

$4 x (x^{2}) - 8 x = 0$

Step 1.2.2.2

Factor $4 x$ out of $- 8 x$ .

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

Step 1.2.2.3

Factor $4 x$ out of $4 x (x^{2}) + 4 x (- 2)$ .

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

Step 1.2.3

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

$x = 0$

$x^{2} - 2 = 0$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

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

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Step 1.2.5.1

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

$x^{2} - 2 = 0$

Step 1.2.5.2

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

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Step 1.2.5.2.1

Add $2$ to both sides of the equation.

$x^{2} = 2$

Step 1.2.5.2.2

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

$x = \pm \sqrt{2}$

Step 1.2.5.2.3

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

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Step 1.2.5.2.3.1

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

$x = \sqrt{2}$

Step 1.2.5.2.3.2

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

$x = - \sqrt{2}$

Step 1.2.5.2.3.3

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

$x = \sqrt{2}, - \sqrt{2}$

Step 1.2.6

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

$x = 0, \sqrt{2}, - \sqrt{2}$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

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 1.4

Evaluate $x^{4} - 4 x^{2}$ 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 $x$ .

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

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

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

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

Step 1.4.1.2.1.2

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

$0 - 4 \cdot 0$

Step 1.4.1.2.1.3

Multiply $- 4$ by $0$ .

$0 + 0$

Step 1.4.1.2.2

Add $0$ and $0$ .

$0$

Step 1.4.2

Evaluate at $x = \sqrt{2}$ .

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Step 1.4.2.1

Substitute $\sqrt{2}$ for $x$ .

${(\sqrt{2})}^{4} - 4 {(\sqrt{2})}^{2}$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Simplify each term.

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Step 1.4.2.2.1.1

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

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Step 1.4.2.2.1.1.1

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

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

Step 1.4.2.2.1.1.2

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

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

Step 1.4.2.2.1.1.3

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

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

Step 1.4.2.2.1.1.4

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

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Step 1.4.2.2.1.1.4.1

Factor $2$ out of $4$ .

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

Step 1.4.2.2.1.1.4.2

Cancel the common factors.

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Step 1.4.2.2.1.1.4.2.1

Factor $2$ out of $2$ .

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

Step 1.4.2.2.1.1.4.2.2

Cancel the common factor.

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

Step 1.4.2.2.1.1.4.2.3

Rewrite the expression.

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

Step 1.4.2.2.1.1.4.2.4

Divide $2$ by $1$ .

$2^{2} - 4 {(\sqrt{2})}^{2}$

Step 1.4.2.2.1.2

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

$4 - 4 {(\sqrt{2})}^{2}$

Step 1.4.2.2.1.3

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

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Step 1.4.2.2.1.3.1

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

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

Step 1.4.2.2.1.3.2

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

$4 - 4 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 1.4.2.2.1.3.3

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

$4 - 4 \cdot 2^{\frac{2}{2}}$

Step 1.4.2.2.1.3.4

Cancel the common factor of $2$ .

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Step 1.4.2.2.1.3.4.1

Cancel the common factor.

$4 - 4 \cdot 2^{\frac{2}{2}}$

Step 1.4.2.2.1.3.4.2

Rewrite the expression.

$4 - 4 \cdot 2^{1}$

Step 1.4.2.2.1.3.5

Evaluate the exponent.

$4 - 4 \cdot 2$

Step 1.4.2.2.1.4

Multiply $- 4$ by $2$ .

$4 - 8$

Step 1.4.2.2.2

Subtract $8$ from $4$ .

$- 4$

Step 1.4.3

Evaluate at $x = - \sqrt{2}$ .

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Step 1.4.3.1

Substitute $- \sqrt{2}$ for $x$ .

${(- \sqrt{2})}^{4} - 4 {(- \sqrt{2})}^{2}$

Step 1.4.3.2

Simplify.

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Step 1.4.3.2.1

Simplify each term.

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Step 1.4.3.2.1.1

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

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

Step 1.4.3.2.1.2

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

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

Step 1.4.3.2.1.3

Multiply ${\sqrt{2}}^{4}$ by $1$ .

${\sqrt{2}}^{4} - 4 {(- \sqrt{2})}^{2}$

Step 1.4.3.2.1.4

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

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Step 1.4.3.2.1.4.1

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

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

Step 1.4.3.2.1.4.2

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

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

Step 1.4.3.2.1.4.3

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

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

Step 1.4.3.2.1.4.4

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

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Step 1.4.3.2.1.4.4.1

Factor $2$ out of $4$ .

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

Step 1.4.3.2.1.4.4.2

Cancel the common factors.

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Step 1.4.3.2.1.4.4.2.1

Factor $2$ out of $2$ .

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

Step 1.4.3.2.1.4.4.2.2

Cancel the common factor.

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

Step 1.4.3.2.1.4.4.2.3

Rewrite the expression.

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

Step 1.4.3.2.1.4.4.2.4

Divide $2$ by $1$ .

$2^{2} - 4 {(- \sqrt{2})}^{2}$

Step 1.4.3.2.1.5

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

$4 - 4 {(- \sqrt{2})}^{2}$

Step 1.4.3.2.1.6

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

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

Step 1.4.3.2.1.7

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

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

Step 1.4.3.2.1.8

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

$4 - 4 {\sqrt{2}}^{2}$

Step 1.4.3.2.1.9

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

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Step 1.4.3.2.1.9.1

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

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

Step 1.4.3.2.1.9.2

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

$4 - 4 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 1.4.3.2.1.9.3

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

$4 - 4 \cdot 2^{\frac{2}{2}}$

Step 1.4.3.2.1.9.4

Cancel the common factor of $2$ .

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Step 1.4.3.2.1.9.4.1

Cancel the common factor.

$4 - 4 \cdot 2^{\frac{2}{2}}$

Step 1.4.3.2.1.9.4.2

Rewrite the expression.

$4 - 4 \cdot 2^{1}$

Step 1.4.3.2.1.9.5

Evaluate the exponent.

$4 - 4 \cdot 2$

Step 1.4.3.2.1.10

Multiply $- 4$ by $2$ .

$4 - 8$

Step 1.4.3.2.2

Subtract $8$ from $4$ .

$- 4$

Step 1.4.4

List all of the points.

$(0, 0), (\sqrt{2}, - 4), (- \sqrt{2}, - 4)$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = - 3$ .

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Step 2.1.1

Substitute $- 3$ for $x$ .

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

Step 2.1.2

Simplify.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Raise $- 3$ to the power of $4$ .

$81 - 4 {(- 3)}^{2}$

Step 2.1.2.1.2

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

$81 - 4 \cdot 9$

Step 2.1.2.1.3

Multiply $- 4$ by $9$ .

$81 - 36$

Step 2.1.2.2

Subtract $36$ from $81$ .

$45$

Step 2.2

Evaluate at $x = 4$ .

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Step 2.2.1

Substitute $4$ for $x$ .

${(4)}^{4} - 4 {(4)}^{2}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

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

$256 - 4 {(4)}^{2}$

Step 2.2.2.1.2

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

$256 - 4 \cdot 16$

Step 2.2.2.1.3

Multiply $- 4$ by $16$ .

$256 - 64$

Step 2.2.2.2

Subtract $64$ from $256$ .

$192$

Step 2.3

List all of the points.

$(- 3, 45), (4, 192)$

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: $(4, 192)$

Absolute Minimum: $(\sqrt{2}, - 4), (- \sqrt{2}, - 4)$

Step 4