Calculus Examples

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

Step 1

Write $x \sqrt{4 - x^{2}}$ as a function.

$f (x) = x \sqrt{4 - x^{2}}$

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - x^{2}}$ as ${(4 - x^{2})}^{\frac{1}{2}}$ .

$f' (x) = \frac{d}{d x} (x {(4 - x^{2})}^{\frac{1}{2}})$

Step 2.1.1.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = {(4 - x^{2})}^{\frac{1}{2}}$ .

$f' (x) = x \frac{d}{d x} ({(4 - x^{2})}^{\frac{1}{2}}) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 4 - x^{2}$ .

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Step 2.1.1.3.1

To apply the Chain Rule, set $u$ as $4 - x^{2}$ .

$f' (x) = x (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (4 - x^{2})) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.3.2

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

$f' (x) = x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (4 - x^{2})) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.3.3

Replace all occurrences of $u$ with $4 - x^{2}$ .

$f' (x) = x (\frac{1}{2} \cdot {(4 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} (4 - x^{2})) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.4

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

$f' (x) = x (\frac{1}{2} \cdot {(4 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (4 - x^{2})) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.5

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

$f' (x) = x (\frac{1}{2} \cdot {(4 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (4 - x^{2})) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.6

Combine the numerators over the common denominator.

$f' (x) = x (\frac{1}{2} \cdot {(4 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (4 - x^{2})) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.7

Simplify the numerator.

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Step 2.1.1.7.1

Multiply $- 1$ by $2$ .

$f' (x) = x (\frac{1}{2} \cdot {(4 - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} (4 - x^{2})) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.7.2

Subtract $2$ from $1$ .

$f' (x) = x (\frac{1}{2} \cdot {(4 - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} (4 - x^{2})) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.8

Combine fractions.

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Step 2.1.1.8.1

Move the negative in front of the fraction.

$f' (x) = x (\frac{1}{2} \cdot {(4 - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} (4 - x^{2})) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.8.2

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

$f' (x) = x (\frac{{(4 - x^{2})}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (4 - x^{2})) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.8.3

Move ${(4 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = x (\frac{1}{2 {(4 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (4 - x^{2})) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.8.4

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

$f' (x) = \frac{x}{2 {(4 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (4 - x^{2}) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.9

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

$f' (x) = \frac{x}{2 {(4 - x^{2})}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (4) + \frac{d}{d x} (- x^{2})) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.10

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$f' (x) = \frac{x}{2 {(4 - x^{2})}^{\frac{1}{2}}} \cdot (0 + \frac{d}{d x} (- x^{2})) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.11

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

$f' (x) = \frac{x}{2 {(4 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x^{2}) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.12

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

$f' (x) = \frac{x}{2 {(4 - x^{2})}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x^{2}) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.13

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

$f' (x) = \frac{x}{2 {(4 - x^{2})}^{\frac{1}{2}}} \cdot (- (2 x)) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.14

Combine fractions.

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Step 2.1.1.14.1

Multiply $2$ by $- 1$ .

$f' (x) = \frac{x}{2 {(4 - x^{2})}^{\frac{1}{2}}} \cdot (- 2 x) + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.14.2

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

$f' (x) = \frac{- 2 x}{2 {(4 - x^{2})}^{\frac{1}{2}}} \cdot x + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.14.3

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

$f' (x) = \frac{- 2 x \cdot x}{2 {(4 - x^{2})}^{\frac{1}{2}}} + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.15

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

$f' (x) = \frac{- 2 (x \cdot x)}{2 {(4 - x^{2})}^{\frac{1}{2}}} + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.16

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

$f' (x) = \frac{- 2 (x \cdot x)}{2 {(4 - x^{2})}^{\frac{1}{2}}} + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.17

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

$f' (x) = \frac{- 2 x^{1 + 1}}{2 {(4 - x^{2})}^{\frac{1}{2}}} + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.18

Add $1$ and $1$ .

$f' (x) = \frac{- 2 x^{2}}{2 {(4 - x^{2})}^{\frac{1}{2}}} + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.19

Factor $2$ out of $- 2 x^{2}$ .

$f' (x) = \frac{2 (- x^{2})}{2 {(4 - x^{2})}^{\frac{1}{2}}} + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.20

Cancel the common factors.

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Step 2.1.1.20.1

Factor $2$ out of $2 {(4 - x^{2})}^{\frac{1}{2}}$ .

$f' (x) = \frac{2 (- x^{2})}{2 ({(4 - x^{2})}^{\frac{1}{2}})} + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.20.2

Cancel the common factor.

$f' (x) = \frac{2 (- x^{2})}{2 {(4 - x^{2})}^{\frac{1}{2}}} + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.20.3

Rewrite the expression.

$f' (x) = \frac{- x^{2}}{{(4 - x^{2})}^{\frac{1}{2}}} + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.21

Move the negative in front of the fraction.

$f' (x) = - \frac{x^{2}}{{(4 - x^{2})}^{\frac{1}{2}}} + {(4 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 2.1.1.22

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

$f' (x) = - \frac{x^{2}}{{(4 - x^{2})}^{\frac{1}{2}}} + {(4 - x^{2})}^{\frac{1}{2}} \cdot 1$

Step 2.1.1.23

Multiply ${(4 - x^{2})}^{\frac{1}{2}}$ by $1$ .

$f' (x) = - \frac{x^{2}}{{(4 - x^{2})}^{\frac{1}{2}}} + {(4 - x^{2})}^{\frac{1}{2}}$

Step 2.1.1.24

To write ${(4 - x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(4 - x^{2})}^{\frac{1}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$ .

$f' (x) = - \frac{x^{2}}{{(4 - x^{2})}^{\frac{1}{2}}} + \frac{{(4 - x^{2})}^{\frac{1}{2}} {(4 - x^{2})}^{\frac{1}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 2.1.1.25

Combine the numerators over the common denominator.

$f' (x) = \frac{- x^{2} + {(4 - x^{2})}^{\frac{1}{2}} {(4 - x^{2})}^{\frac{1}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 2.1.1.26

Multiply ${(4 - x^{2})}^{\frac{1}{2}}$ by ${(4 - x^{2})}^{\frac{1}{2}}$ by adding the exponents.

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Step 2.1.1.26.1

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

$f' (x) = \frac{- x^{2} + {(4 - x^{2})}^{\frac{1}{2} + \frac{1}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 2.1.1.26.2

Combine the numerators over the common denominator.

$f' (x) = \frac{- x^{2} + {(4 - x^{2})}^{\frac{1 + 1}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 2.1.1.26.3

Add $1$ and $1$ .

$f' (x) = \frac{- x^{2} + {(4 - x^{2})}^{\frac{2}{2}}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 2.1.1.26.4

Divide $2$ by $2$ .

$f' (x) = \frac{- x^{2} + 4 - x^{2}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 2.1.1.27

Simplify ${(4 - x^{2})}^{1}$ .

$f' (x) = \frac{- x^{2} + 4 - x^{2}}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 2.1.1.28

Subtract $x^{2}$ from $- x^{2}$ .

$f' (x) = \frac{- 2 x^{2} + 4}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 2.1.1.29

Reorder terms.

$f' (x) = \frac{- 2 x^{2} + 4}{{(- x^{2} + 4)}^{\frac{1}{2}}}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = - 2 x^{2} + 4$ and $g (x) = {(- x^{2} + 4)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} \frac{d}{d x} (- 2 x^{2} + 4) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{{({(- x^{2} + 4)}^{\frac{1}{2}})}^{2}}$

Step 2.1.2.2

Multiply the exponents in ${({(- x^{2} + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 2.1.2.2.1

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} \frac{d}{d x} (- 2 x^{2} + 4) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{{(- x^{2} + 4)}^{\frac{1}{2} \cdot 2}}$

Step 2.1.2.2.2

Cancel the common factor of $2$ .

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Step 2.1.2.2.2.1

Cancel the common factor.

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} \frac{d}{d x} (- 2 x^{2} + 4) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{{(- x^{2} + 4)}^{\frac{1}{2} \cdot 2}}$

Step 2.1.2.2.2.2

Rewrite the expression.

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} \frac{d}{d x} (- 2 x^{2} + 4) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{- x^{2} + 4}$

Step 2.1.2.3

Simplify.

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} \frac{d}{d x} (- 2 x^{2} + 4) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{- x^{2} + 4}$

Step 2.1.2.4

Differentiate.

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Step 2.1.2.4.1

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (\frac{d}{d x} (- 2 x^{2}) + \frac{d}{d x} (4)) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{- x^{2} + 4}$

Step 2.1.2.4.2

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 2 \frac{d}{d x} x^{2} + \frac{d}{d x} (4)) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{- x^{2} + 4}$

Step 2.1.2.4.3

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 2 (2 x) + \frac{d}{d x} (4)) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{- x^{2} + 4}$

Step 2.1.2.4.4

Multiply $2$ by $- 2$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x + \frac{d}{d x} (4)) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{- x^{2} + 4}$

Step 2.1.2.4.5

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x + 0) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{- x^{2} + 4}$

Step 2.1.2.4.6

Add $- 4 x$ and $0$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 4)}^{\frac{1}{2}}}{- x^{2} + 4}$

Step 2.1.2.5

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

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Step 2.1.2.5.1

To apply the Chain Rule, set $u$ as $- x^{2} + 4$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 2.1.2.5.2

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 2.1.2.5.3

Replace all occurrences of $u$ with $- x^{2} + 4$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 4)}^{\frac{1}{2} - 1} \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 2.1.2.6

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 4)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 2.1.2.7

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 4)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 2.1.2.8

Combine the numerators over the common denominator.

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 4)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 2.1.2.9

Simplify the numerator.

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Step 2.1.2.9.1

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 4)}^{\frac{1 - 2}{2}} \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 2.1.2.9.2

Subtract $2$ from $1$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 4)}^{\frac{- 1}{2}} \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 2.1.2.10

Combine fractions.

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Step 2.1.2.10.1

Move the negative in front of the fraction.

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 4)}^{- \frac{1}{2}} \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 2.1.2.10.2

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{{(- x^{2} + 4)}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 2.1.2.10.3

Move ${(- x^{2} + 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 4)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x^{2} + 4))}{- x^{2} + 4}$

Step 2.1.2.11

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 4)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (- x^{2}) + \frac{d}{d x} (4)))}{- x^{2} + 4}$

Step 2.1.2.12

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 4)}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x^{2} + \frac{d}{d x} (4)))}{- x^{2} + 4}$

Step 2.1.2.13

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 4)}^{\frac{1}{2}}} \cdot (- (2 x) + \frac{d}{d x} (4)))}{- x^{2} + 4}$

Step 2.1.2.14

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 4)}^{\frac{1}{2}}} \cdot (- 2 x + \frac{d}{d x} (4)))}{- x^{2} + 4}$

Step 2.1.2.15

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 4)}^{\frac{1}{2}}} \cdot (- 2 x + 0))}{- x^{2} + 4}$

Step 2.1.2.16

Simplify terms.

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Step 2.1.2.16.1

Add $- 2 x$ and $0$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 4)}^{\frac{1}{2}}} \cdot (- 2 x))}{- x^{2} + 4}$

Step 2.1.2.16.2

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (\frac{- 2}{2 {(- x^{2} + 4)}^{\frac{1}{2}}} \cdot x)}{- x^{2} + 4}$

Step 2.1.2.16.3

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

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) \frac{- 2 x}{2 {(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.16.4

Factor $2$ out of $- 2 x$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) \frac{2 (- x)}{2 {(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.17

Cancel the common factors.

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Step 2.1.2.17.1

Factor $2$ out of $2 {(- x^{2} + 4)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) \frac{2 (- x)}{2 ({(- x^{2} + 4)}^{\frac{1}{2}})}}{- x^{2} + 4}$

Step 2.1.2.17.2

Cancel the common factor.

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) \frac{2 (- x)}{2 {(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.17.3

Rewrite the expression.

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) \frac{- x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.18

Move the negative in front of the fraction.

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 4) (- \frac{x}{{(- x^{2} + 4)}^{\frac{1}{2}}})}{- x^{2} + 4}$

Step 2.1.2.19

Multiply $- 1$ by $- 1$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) + 1 (- 2 x^{2} + 4) (\frac{x}{{(- x^{2} + 4)}^{\frac{1}{2}}})}{- x^{2} + 4}$

Step 2.1.2.20

Multiply $- 2 x^{2} + 4$ by $1$ .

$f'' (x) = \frac{{(- x^{2} + 4)}^{\frac{1}{2}} (- 4 x) + (- 2 x^{2} + 4) (\frac{x}{{(- x^{2} + 4)}^{\frac{1}{2}}})}{- x^{2} + 4}$

Step 2.1.2.21

Simplify.

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Step 2.1.2.21.1

Simplify the numerator.

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Step 2.1.2.21.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x + (- 2 x^{2} + 4) (\frac{x}{{(- x^{2} + 4)}^{\frac{1}{2}}})}{- x^{2} + 4}$

Step 2.1.2.21.1.2

Multiply $- 2 x^{2} + 4$ by $\frac{x}{{(- x^{2} + 4)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x + \frac{(- 2 x^{2} + 4) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.3

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

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Step 2.1.2.21.1.3.1

Factor $2$ out of $- 2 x^{2}$ .

$f'' (x) = \frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x + \frac{(2 (- x^{2}) + 4) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.3.2

Factor $2$ out of $4$ .

$f'' (x) = \frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x + \frac{(2 (- x^{2}) + 2 (2)) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.3.3

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

$f'' (x) = \frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x + \frac{2 (- x^{2} + 2) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.4

To write $- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x$ as a fraction with a common denominator, multiply by $\frac{{(- x^{2} + 4)}^{\frac{1}{2}}}{{(- x^{2} + 4)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x \cdot \frac{{(- x^{2} + 4)}^{\frac{1}{2}}}{{(- x^{2} + 4)}^{\frac{1}{2}}} + \frac{2 (- x^{2} + 2) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.5

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

$f'' (x) = \frac{\frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x {(- x^{2} + 4)}^{\frac{1}{2}}}{{(- x^{2} + 4)}^{\frac{1}{2}}} + \frac{2 (- x^{2} + 2) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.6

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x {(- x^{2} + 4)}^{\frac{1}{2}} + 2 (- x^{2} + 2) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.7

Rewrite $\frac{- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x {(- x^{2} + 4)}^{\frac{1}{2}} + 2 (- x^{2} + 2) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}$ in a factored form.

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Step 2.1.2.21.1.7.1

Factor $2 x$ out of $- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x {(- x^{2} + 4)}^{\frac{1}{2}} + 2 (- x^{2} + 2) x$ .

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Step 2.1.2.21.1.7.1.1

Factor $2 x$ out of $- 4 {(- x^{2} + 4)}^{\frac{1}{2}} x {(- x^{2} + 4)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 4)}^{\frac{1}{2}} {(- x^{2} + 4)}^{\frac{1}{2}}) + 2 (- x^{2} + 2) x}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.7.1.2

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

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 4)}^{\frac{1}{2}} {(- x^{2} + 4)}^{\frac{1}{2}}) + 2 x (- x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.7.1.3

Factor $2 x$ out of $2 x (- 2 {(- x^{2} + 4)}^{\frac{1}{2}} {(- x^{2} + 4)}^{\frac{1}{2}}) + 2 x (- x^{2} + 2)$ .

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 4)}^{\frac{1}{2}} {(- x^{2} + 4)}^{\frac{1}{2}} - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.7.2

Combine exponents.

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Step 2.1.2.21.1.7.2.1

Multiply ${(- x^{2} + 4)}^{\frac{1}{2}}$ by ${(- x^{2} + 4)}^{\frac{1}{2}}$ by adding the exponents.

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Step 2.1.2.21.1.7.2.1.1

Move ${(- x^{2} + 4)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{\frac{2 x (- 2 ({(- x^{2} + 4)}^{\frac{1}{2}} {(- x^{2} + 4)}^{\frac{1}{2}}) - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.7.2.1.2

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

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 4)}^{\frac{1}{2} + \frac{1}{2}} - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.7.2.1.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 4)}^{\frac{1 + 1}{2}} - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.7.2.1.4

Add $1$ and $1$ .

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 4)}^{\frac{2}{2}} - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.7.2.1.5

Divide $2$ by $2$ .

$f'' (x) = \frac{\frac{2 x (- 2 (- x^{2} + 4) - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.7.2.2

Simplify $- 2 {(- x^{2} + 4)}^{1}$ .

$f'' (x) = \frac{\frac{2 x (- 2 (- x^{2} + 4) - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.8

Simplify the numerator.

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Step 2.1.2.21.1.8.1

Apply the distributive property.

$f'' (x) = \frac{\frac{2 x (- 2 (- x^{2}) - 2 \cdot 4 - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.8.2

Multiply $- 1$ by $- 2$ .

$f'' (x) = \frac{\frac{2 x (2 x^{2} - 2 \cdot 4 - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.8.3

Multiply $- 2$ by $4$ .

$f'' (x) = \frac{\frac{2 x (2 x^{2} - 8 - x^{2} + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.8.4

Subtract $x^{2}$ from $2 x^{2}$ .

$f'' (x) = \frac{\frac{2 x (x^{2} - 8 + 2)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.1.8.5

Add $- 8$ and $2$ .

$f'' (x) = \frac{\frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$

Step 2.1.2.21.2

Combine terms.

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Step 2.1.2.21.2.1

Rewrite $\frac{\frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1}{2}}}}{- x^{2} + 4}$ as a product.

$f'' (x) = \frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1}{2}}} \cdot \frac{1}{- x^{2} + 4}$

Step 2.1.2.21.2.2

Multiply $\frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1}{2}}}$ by $\frac{1}{- x^{2} + 4}$ .

$f'' (x) = \frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1}{2}} (- x^{2} + 4)}$

Step 2.1.2.21.2.3

Multiply ${(- x^{2} + 4)}^{\frac{1}{2}}$ by $- x^{2} + 4$ by adding the exponents.

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Step 2.1.2.21.2.3.1

Multiply ${(- x^{2} + 4)}^{\frac{1}{2}}$ by $- x^{2} + 4$ .

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Step 2.1.2.21.2.3.1.1

Raise $- x^{2} + 4$ to the power of $1$ .

$f'' (x) = \frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1}{2}} (- x^{2} + 4)}$

Step 2.1.2.21.2.3.1.2

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

$f'' (x) = \frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1}{2} + 1}}$

Step 2.1.2.21.2.3.2

Write $1$ as a fraction with a common denominator.

$f'' (x) = \frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1}{2} + \frac{2}{2}}}$

Step 2.1.2.21.2.3.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{1 + 2}{2}}}$

Step 2.1.2.21.2.3.4

Add $1$ and $2$ .

$f'' (x) = \frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{3}{2}}}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{3}{2}}}$ .

$\frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{3}{2}}}$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $\frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{3}{2}}} = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$\frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{3}{2}}} = 0$

Step 2.2.2

Set the numerator equal to zero.

$2 x (x^{2} - 6) = 0$

Step 2.2.3

Solve the equation for $x$ .

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Step 2.2.3.1

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

$x = 0$

$x^{2} - 6 = 0$

Step 2.2.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 2.2.3.3

Set $x^{2} - 6$ equal to $0$ and solve for $x$ .

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Step 2.2.3.3.1

Set $x^{2} - 6$ equal to $0$ .

$x^{2} - 6 = 0$

Step 2.2.3.3.2

Solve $x^{2} - 6 = 0$ for $x$ .

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Step 2.2.3.3.2.1

Add $6$ to both sides of the equation.

$x^{2} = 6$

Step 2.2.3.3.2.2

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

$x = \pm \sqrt{6}$

Step 2.2.3.3.2.3

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

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Step 2.2.3.3.2.3.1

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

$x = \sqrt{6}$

Step 2.2.3.3.2.3.2

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

$x = - \sqrt{6}$

Step 2.2.3.3.2.3.3

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

$x = \sqrt{6}, - \sqrt{6}$

Step 2.2.3.4

The final solution is all the values that make $2 x (x^{2} - 6) = 0$ true.

$x = 0, \sqrt{6}, - \sqrt{6}$

Step 2.2.4

Exclude the solutions that do not make $\frac{2 x (x^{2} - 6)}{{(- x^{2} + 4)}^{\frac{3}{2}}} = 0$ true.

$x = 0$

Step 3

Find the domain of $f (x) = x \sqrt{4 - x^{2}}$ .

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Step 3.1

Set the radicand in $\sqrt{4 - x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$4 - x^{2} \geq 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Subtract $4$ from both sides of the inequality.

$- x^{2} \geq - 4$

Step 3.2.2

Divide each term in $- x^{2} \geq - 4$ by $- 1$ and simplify.

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Step 3.2.2.1

Divide each term in $- x^{2} \geq - 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 4}{- 1}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 4}{- 1}$

Step 3.2.2.2.2

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

$x^{2} \leq \frac{- 4}{- 1}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $- 4$ by $- 1$ .

$x^{2} \leq 4$

Step 3.2.3

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

$\sqrt{x^{2}} \leq \sqrt{4}$

Step 3.2.4

Simplify the equation.

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Step 3.2.4.1

Simplify the left side.

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Step 3.2.4.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{4}$

Step 3.2.4.2

Simplify the right side.

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Step 3.2.4.2.1

Simplify $\sqrt{4}$ .

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Step 3.2.4.2.1.1

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

$| x | \leq \sqrt{2^{2}}$

Step 3.2.4.2.1.2

Pull terms out from under the radical.

$| x | \leq | 2 |$

Step 3.2.4.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$| x | \leq 2$

Step 3.2.5

Write $| x | \leq 2$ as a piecewise.

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Step 3.2.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 3.2.5.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq 2$

Step 3.2.5.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 3.2.5.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq 2$

Step 3.2.5.5

Write as a piecewise.

${\begin{cases} x \leq 2 & x \geq 0 \\ - x \leq 2 & x < 0 \end{cases}$

Step 3.2.6

Find the intersection of $x \leq 2$ and $x \geq 0$ .

$0 \leq x \leq 2$

Step 3.2.7

Solve $- x \leq 2$ when $x < 0$ .

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Step 3.2.7.1

Divide each term in $- x \leq 2$ by $- 1$ and simplify.

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Step 3.2.7.1.1

Divide each term in $- x \leq 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{2}{- 1}$

Step 3.2.7.1.2

Simplify the left side.

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Step 3.2.7.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{2}{- 1}$

Step 3.2.7.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{2}{- 1}$

Step 3.2.7.1.3

Simplify the right side.

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Step 3.2.7.1.3.1

Divide $2$ by $- 1$ .

$x \geq - 2$

Step 3.2.7.2

Find the intersection of $x \geq - 2$ and $x < 0$ .

$- 2 \leq x < 0$

Step 3.2.8

Find the union of the solutions.

$- 2 \leq x \leq 2$

Step 3.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[- 2, 2]$

Set-Builder Notation:

${x | - 2 \leq x \leq 2}$

Interval Notation:

$[- 2, 2]$

Set-Builder Notation:

${x | - 2 \leq x \leq 2}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$[- 2, 0) \cup (0, 2]$

Step 5

Substitute any number from the interval $[- 2, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

$f'' (- 1) = \frac{2 (- 1) ({(- 1)}^{2} - 6)}{{(- {(- 1)}^{2} + 4)}^{\frac{3}{2}}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $2$ by $- 1$ .

$f'' (- 1) = \frac{- 2 ({(- 1)}^{2} - 6)}{{(- {(- 1)}^{2} + 4)}^{\frac{3}{2}}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Simplify each term.

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Step 5.2.2.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 5.2.2.1.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 5.2.2.1.1.1.1

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

$f'' (- 1) = \frac{- 2 ({(- 1)}^{2} - 6)}{{((- 1) {(- 1)}^{2} + 4)}^{\frac{3}{2}}}$

Step 5.2.2.1.1.1.2

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

$f'' (- 1) = \frac{- 2 ({(- 1)}^{2} - 6)}{{({(- 1)}^{1 + 2} + 4)}^{\frac{3}{2}}}$

Step 5.2.2.1.1.2

Add $1$ and $2$ .

$f'' (- 1) = \frac{- 2 ({(- 1)}^{2} - 6)}{{({(- 1)}^{3} + 4)}^{\frac{3}{2}}}$

Step 5.2.2.1.2

Raise $- 1$ to the power of $3$ .

$f'' (- 1) = \frac{- 2 ({(- 1)}^{2} - 6)}{{(- 1 + 4)}^{\frac{3}{2}}}$

Step 5.2.2.2

Add $- 1$ and $4$ .

$f'' (- 1) = \frac{- 2 ({(- 1)}^{2} - 6)}{3^{\frac{3}{2}}}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

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

$f'' (- 1) = \frac{- 2 (1 - 6)}{3^{\frac{3}{2}}}$

Step 5.2.3.2

Subtract $6$ from $1$ .

$f'' (- 1) = \frac{- 2 \cdot - 5}{3^{\frac{3}{2}}}$

Step 5.2.4

Multiply $- 2$ by $- 5$ .

$f'' (- 1) = \frac{10}{3^{\frac{3}{2}}}$

Step 5.2.5

The final answer is $\frac{10}{3^{\frac{3}{2}}}$ .

$\frac{10}{3^{\frac{3}{2}}}$

Step 5.3

The graph is concave up on the interval $[- 2, 0)$ because $f'' (- 1)$ is positive.

Concave up on $[- 2, 0)$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(0, 2]$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

$f'' (1) = \frac{2 (1) ({(1)}^{2} - 6)}{{(- {(1)}^{2} + 4)}^{\frac{3}{2}}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $2$ by $1$ .

$f'' (1) = \frac{2 (1^{2} - 6)}{{(- 1^{2} + 4)}^{\frac{3}{2}}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

One to any power is one.

$f'' (1) = \frac{2 (1^{2} - 6)}{{(- 1 \cdot 1 + 4)}^{\frac{3}{2}}}$

Step 6.2.2.1.2

Multiply $- 1$ by $1$ .

$f'' (1) = \frac{2 (1^{2} - 6)}{{(- 1 + 4)}^{\frac{3}{2}}}$

Step 6.2.2.2

Add $- 1$ and $4$ .

$f'' (1) = \frac{2 (1^{2} - 6)}{3^{\frac{3}{2}}}$

Step 6.2.3

Simplify the numerator.

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Step 6.2.3.1

One to any power is one.

$f'' (1) = \frac{2 (1 - 6)}{3^{\frac{3}{2}}}$

Step 6.2.3.2

Subtract $6$ from $1$ .

$f'' (1) = \frac{2 \cdot - 5}{3^{\frac{3}{2}}}$

Step 6.2.4

Simplify the expression.

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Step 6.2.4.1

Multiply $2$ by $- 5$ .

$f'' (1) = \frac{- 10}{3^{\frac{3}{2}}}$

Step 6.2.4.2

Move the negative in front of the fraction.

$f'' (1) = - \frac{10}{3^{\frac{3}{2}}}$

Step 6.2.5

The final answer is $- \frac{10}{3^{\frac{3}{2}}}$ .

$- \frac{10}{3^{\frac{3}{2}}}$

Step 6.3

The graph is concave down on the interval $(0, 2]$ because $f'' (1)$ is negative.

Concave down on $(0, 2]$ since $f'' (x)$ is negative

Step 7

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave up on $[- 2, 0)$ since $f'' (x)$ is positive

Concave down on $(0, 2]$ since $f'' (x)$ is negative

Step 8