Calculus Examples

$f (x) = \sqrt{x^{3} + 8 x}$

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3} + 8 x}$ as ${(x^{3} + 8 x)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(x^{3} + 8 x)}^{\frac{1}{2}}]$

Step 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 $f (x) = x^{\frac{1}{2}}$ and $g (x) = x^{3} + 8 x$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $x^{3} + 8 x$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{3} + 8 x]$

Step 1.1.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{3} + 8 x]$

Step 1.1.2.3

Replace all occurrences of $u$ with $x^{3} + 8 x$ .

$\frac{1}{2} {(x^{3} + 8 x)}^{\frac{1}{2} - 1} \frac{d}{d x} [x^{3} + 8 x]$

Step 1.1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} {(x^{3} + 8 x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{3} + 8 x]$

Step 1.1.4

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

$\frac{1}{2} {(x^{3} + 8 x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{3} + 8 x]$

Step 1.1.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(x^{3} + 8 x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{3} + 8 x]$

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(x^{3} + 8 x)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x^{3} + 8 x]$

Step 1.1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(x^{3} + 8 x)}^{\frac{- 1}{2}} \frac{d}{d x} [x^{3} + 8 x]$

Step 1.1.7

Combine fractions.

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Step 1.1.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(x^{3} + 8 x)}^{- \frac{1}{2}} \frac{d}{d x} [x^{3} + 8 x]$

Step 1.1.7.2

Combine $\frac{1}{2}$ and ${(x^{3} + 8 x)}^{- \frac{1}{2}}$ .

$\frac{{(x^{3} + 8 x)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x^{3} + 8 x]$

Step 1.1.7.3

Move ${(x^{3} + 8 x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(x^{3} + 8 x)}^{\frac{1}{2}}} \frac{d}{d x} [x^{3} + 8 x]$

Step 1.1.8

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

$\frac{1}{2 {(x^{3} + 8 x)}^{\frac{1}{2}}} (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [8 x])$

Step 1.1.9

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

$\frac{1}{2 {(x^{3} + 8 x)}^{\frac{1}{2}}} (3 x^{2} + \frac{d}{d x} [8 x])$

Step 1.1.10

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

$\frac{1}{2 {(x^{3} + 8 x)}^{\frac{1}{2}}} (3 x^{2} + 8 \frac{d}{d x} [x])$

Step 1.1.11

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

$\frac{1}{2 {(x^{3} + 8 x)}^{\frac{1}{2}}} (3 x^{2} + 8 \cdot 1)$

Step 1.1.12

Multiply $8$ by $1$ .

$\frac{1}{2 {(x^{3} + 8 x)}^{\frac{1}{2}}} (3 x^{2} + 8)$

Step 1.1.13

Simplify.

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Step 1.1.13.1

Reorder the factors of $\frac{1}{2 {(x^{3} + 8 x)}^{\frac{1}{2}}} (3 x^{2} + 8)$ .

$(3 x^{2} + 8) \frac{1}{2 {(x^{3} + 8 x)}^{\frac{1}{2}}}$

Step 1.1.13.2

Multiply $3 x^{2} + 8$ by $\frac{1}{2 {(x^{3} + 8 x)}^{\frac{1}{2}}}$ .

$f' (x) = \frac{3 x^{2} + 8}{2 {(x^{3} + 8 x)}^{\frac{1}{2}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{3 x^{2} + 8}{2 {(x^{3} + 8 x)}^{\frac{1}{2}}}$ .

$\frac{3 x^{2} + 8}{2 {(x^{3} + 8 x)}^{\frac{1}{2}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{3 x^{2} + 8}{2 {(x^{3} + 8 x)}^{\frac{1}{2}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{3 x^{2} + 8}{2 {(x^{3} + 8 x)}^{\frac{1}{2}}} = 0$

Step 2.2

Set the numerator equal to zero.

$3 x^{2} + 8 = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Subtract $8$ from both sides of the equation.

$3 x^{2} = - 8$

Step 2.3.2

Divide each term in $3 x^{2} = - 8$ by $3$ and simplify.

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Step 2.3.2.1

Divide each term in $3 x^{2} = - 8$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{- 8}{3}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{- 8}{3}$

Step 2.3.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 8}{3}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{8}{3}$

Step 2.3.3

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

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

Step 2.3.4

Simplify $\pm \sqrt{- \frac{8}{3}}$ .

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Step 2.3.4.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{8}{3}}$

Step 2.3.4.2

Pull terms out from under the radical.

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

Step 2.3.4.3

Rewrite $\sqrt{\frac{8}{3}}$ as $\frac{\sqrt{8}}{\sqrt{3}}$ .

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

Step 2.3.4.4

Simplify the numerator.

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Step 2.3.4.4.1

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

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Step 2.3.4.4.1.1

Factor $4$ out of $8$ .

$x = \pm i \frac{\sqrt{4 (2)}}{\sqrt{3}}$

Step 2.3.4.4.1.2

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

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

Step 2.3.4.4.2

Pull terms out from under the radical.

$x = \pm i \frac{2 \sqrt{2}}{\sqrt{3}}$

Step 2.3.4.5

Multiply $\frac{2 \sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 2.3.4.6

Combine and simplify the denominator.

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Step 2.3.4.6.1

Multiply $\frac{2 \sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i \frac{2 \sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.3.4.6.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm i \frac{2 \sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.3.4.6.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm i \frac{2 \sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.3.4.6.4

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

$x = \pm i \frac{2 \sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.3.4.6.5

Add $1$ and $1$ .

$x = \pm i \frac{2 \sqrt{2} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.3.4.6.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.3.4.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm i \frac{2 \sqrt{2} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.3.4.6.6.2

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

$x = \pm i \frac{2 \sqrt{2} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.3.4.6.6.3

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

$x = \pm i \frac{2 \sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.3.4.6.6.4

Cancel the common factor of $2$ .

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Step 2.3.4.6.6.4.1

Cancel the common factor.

$x = \pm i \frac{2 \sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.3.4.6.6.4.2

Rewrite the expression.

$x = \pm i \frac{2 \sqrt{2} \sqrt{3}}{3^{1}}$

Step 2.3.4.6.6.5

Evaluate the exponent.

$x = \pm i \frac{2 \sqrt{2} \sqrt{3}}{3}$

Step 2.3.4.7

Simplify the numerator.

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Step 2.3.4.7.1

Combine using the product rule for radicals.

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

Step 2.3.4.7.2

Multiply $3$ by $2$ .

$x = \pm i \frac{2 \sqrt{6}}{3}$

Step 2.3.4.8

Combine fractions.

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Step 2.3.4.8.1

Combine $i$ and $\frac{2 \sqrt{6}}{3}$ .

$x = \pm \frac{i (2 \sqrt{6})}{3}$

Step 2.3.4.8.2

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

$x = \pm \frac{2 i \sqrt{6}}{3}$

Step 2.3.5

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

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Step 2.3.5.1

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

$x = \frac{2 i \sqrt{6}}{3}$

Step 2.3.5.2

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

$x = - \frac{2 i \sqrt{6}}{3}$

Step 2.3.5.3

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

$x = \frac{2 i \sqrt{6}}{3}, - \frac{2 i \sqrt{6}}{3}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Convert expressions with fractional exponents to radicals.

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Step 3.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{3 x^{2} + 8}{2 \sqrt{{(x^{3} + 8 x)}^{1}}}$

Step 3.1.2

Anything raised to $1$ is the base itself.

$\frac{3 x^{2} + 8}{2 \sqrt{x^{3} + 8 x}}$

Step 3.2

Set the denominator in $\frac{3 x^{2} + 8}{2 \sqrt{x^{3} + 8 x}}$ equal to $0$ to find where the expression is undefined.

$2 \sqrt{x^{3} + 8 x} = 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${(2 \sqrt{x^{3} + 8 x})}^{2} = 0^{2}$

Step 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3} + 8 x}$ as ${(x^{3} + 8 x)}^{\frac{1}{2}}$ .

${(2 {(x^{3} + 8 x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Simplify ${(2 {(x^{3} + 8 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.2.1.1

Apply the product rule to $2 {(x^{3} + 8 x)}^{\frac{1}{2}}$ .

$2^{2} {({(x^{3} + 8 x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.3.2.2.1.2

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

$4 {({(x^{3} + 8 x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.3.2.2.1.3

Multiply the exponents in ${({(x^{3} + 8 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.2.1.3.1

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

$4 {(x^{3} + 8 x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.3.2.1

Cancel the common factor.

$4 {(x^{3} + 8 x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

$4 {(x^{3} + 8 x)}^{1} = 0^{2}$

Step 3.3.2.2.1.4

Simplify.

$4 (x^{3} + 8 x) = 0^{2}$

Step 3.3.2.2.1.5

Apply the distributive property.

$4 x^{3} + 4 (8 x) = 0^{2}$

Step 3.3.2.2.1.6

Multiply $8$ by $4$ .

$4 x^{3} + 32 x = 0^{2}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

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

$4 x^{3} + 32 x = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Factor $4 x$ out of $4 x^{3} + 32 x$ .

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Step 3.3.3.1.1

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

$4 x (x^{2}) + 32 x = 0$

Step 3.3.3.1.2

Factor $4 x$ out of $32 x$ .

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

Step 3.3.3.1.3

Factor $4 x$ out of $4 x (x^{2}) + 4 x (8)$ .

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

Step 3.3.3.2

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

$x = 0$

$x^{2} + 8 = 0$

Step 3.3.3.3

Set $x$ equal to $0$ .

$x = 0$

Step 3.3.3.4

Set $x^{2} + 8$ equal to $0$ and solve for $x$ .

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Step 3.3.3.4.1

Set $x^{2} + 8$ equal to $0$ .

$x^{2} + 8 = 0$

Step 3.3.3.4.2

Solve $x^{2} + 8 = 0$ for $x$ .

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Step 3.3.3.4.2.1

Subtract $8$ from both sides of the equation.

$x^{2} = - 8$

Step 3.3.3.4.2.2

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

$x = \pm \sqrt{- 8}$

Step 3.3.3.4.2.3

Simplify $\pm \sqrt{- 8}$ .

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Step 3.3.3.4.2.3.1

Rewrite $- 8$ as $- 1 (8)$ .

$x = \pm \sqrt{- 1 (8)}$

Step 3.3.3.4.2.3.2

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{8}$

Step 3.3.3.4.2.3.3

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

$x = \pm i \cdot \sqrt{8}$

Step 3.3.3.4.2.3.4

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

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Step 3.3.3.4.2.3.4.1

Factor $4$ out of $8$ .

$x = \pm i \cdot \sqrt{4 (2)}$

Step 3.3.3.4.2.3.4.2

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

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

Step 3.3.3.4.2.3.5

Pull terms out from under the radical.

$x = \pm i \cdot (2 \sqrt{2})$

Step 3.3.3.4.2.3.6

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

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

Step 3.3.3.4.2.4

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

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Step 3.3.3.4.2.4.1

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

$x = 2 i \sqrt{2}$

Step 3.3.3.4.2.4.2

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

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

Step 3.3.3.4.2.4.3

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

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

Step 3.3.3.5

The final solution is all the values that make $4 x (x^{2} + 8) = 0$ true.

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

Step 3.4

Set the radicand in $\sqrt{x^{3} + 8 x}$ less than $0$ to find where the expression is undefined.

$x^{3} + 8 x < 0$

Step 3.5

Solve for $x$ .

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Step 3.5.1

Convert the inequality to an equation.

$x^{3} + 8 x = 0$

Step 3.5.2

Factor $x$ out of $x^{3} + 8 x$ .

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Step 3.5.2.1

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

$x \cdot x^{2} + 8 x = 0$

Step 3.5.2.2

Factor $x$ out of $8 x$ .

$x \cdot x^{2} + x \cdot 8 = 0$

Step 3.5.2.3

Factor $x$ out of $x \cdot x^{2} + x \cdot 8$ .

$x (x^{2} + 8) = 0$

Step 3.5.3

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

$x = 0$

$x^{2} + 8 = 0$

Step 3.5.4

Set $x$ equal to $0$ .

$x = 0$

Step 3.5.5

Set $x^{2} + 8$ equal to $0$ and solve for $x$ .

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Step 3.5.5.1

Set $x^{2} + 8$ equal to $0$ .

$x^{2} + 8 = 0$

Step 3.5.5.2

Solve $x^{2} + 8 = 0$ for $x$ .

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Step 3.5.5.2.1

Subtract $8$ from both sides of the equation.

$x^{2} = - 8$

Step 3.5.5.2.2

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

$x = \pm \sqrt{- 8}$

Step 3.5.5.2.3

Simplify $\pm \sqrt{- 8}$ .

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Step 3.5.5.2.3.1

Rewrite $- 8$ as $- 1 (8)$ .

$x = \pm \sqrt{- 1 (8)}$

Step 3.5.5.2.3.2

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{8}$

Step 3.5.5.2.3.3

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

$x = \pm i \cdot \sqrt{8}$

Step 3.5.5.2.3.4

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

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Step 3.5.5.2.3.4.1

Factor $4$ out of $8$ .

$x = \pm i \cdot \sqrt{4 (2)}$

Step 3.5.5.2.3.4.2

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

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

Step 3.5.5.2.3.5

Pull terms out from under the radical.

$x = \pm i \cdot (2 \sqrt{2})$

Step 3.5.5.2.3.6

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

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

Step 3.5.5.2.4

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

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Step 3.5.5.2.4.1

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

$x = 2 i \sqrt{2}$

Step 3.5.5.2.4.2

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

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

Step 3.5.5.2.4.3

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

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

Step 3.5.6

The final solution is all the values that make $x (x^{2} + 8) = 0$ true.

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

Step 3.5.7

The solution consists of all of the true intervals.

$x < 0$

Step 3.6

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 $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 4

Evaluate $\sqrt{x^{3} + 8 x}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$\sqrt{{(0)}^{3} + 8 (0)}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

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

$\sqrt{0 + 8 (0)}$

Step 4.1.2.2

Multiply $8$ by $0$ .

$\sqrt{0 + 0}$

Step 4.1.2.3

Add $0$ and $0$ .

$\sqrt{0}$

Step 4.1.2.4

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

$\sqrt{0^{2}}$

Step 4.1.2.5

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

$0$

Step 4.2

List all of the points.

$(0, 0)$

Step 5