Calculus Examples

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

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

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 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 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 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 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 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 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 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 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 1.1.7

Simplify the numerator.

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Step 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 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 1.1.8

Combine fractions.

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Step 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 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 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 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 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 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 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 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 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 1.1.14

Combine fractions.

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Step 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 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 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 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 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 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 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 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 1.1.20

Cancel the common factors.

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Step 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 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 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 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 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 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 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 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 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 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 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 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 1.1.26.4

Divide $2$ by $2$ .

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

Step 1.1.27

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

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

Step 1.1.28

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

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

Step 1.1.29

Reorder terms.

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

Step 1.2

Find the second derivative.

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Step 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 1.2.2

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

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Step 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 1.2.2.2

Cancel the common factor of $2$ .

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Step 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 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 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 1.2.4

Differentiate.

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Step 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 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 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 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 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 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 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 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 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 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 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 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 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 1.2.9

Simplify the numerator.

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Step 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 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 1.2.10

Combine fractions.

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Step 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 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 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 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 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 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 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 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 1.2.16

Simplify terms.

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Step 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 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 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 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 1.2.17

Cancel the common factors.

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Step 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 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 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 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 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 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 1.2.21

Simplify.

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Step 1.2.21.1

Simplify the numerator.

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Step 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 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 1.2.21.1.3

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

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Step 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 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 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 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 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 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 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 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 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 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 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 1.2.21.1.7.2

Combine exponents.

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Step 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 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 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 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 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 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 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 1.2.21.1.8

Simplify the numerator.

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Step 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 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 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 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 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 1.2.21.2

Combine terms.

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Step 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 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 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 1.2.21.2.3.1

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

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Step 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 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 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 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 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 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

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 2.1

Set the second derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

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

Step 2.3

Solve the equation for $x$ .

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Step 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 2.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 2.3.3

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

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Step 2.3.3.1

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

$x^{2} - 6 = 0$

Step 2.3.3.2

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

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Step 2.3.3.2.1

Add $6$ to both sides of the equation.

$x^{2} = 6$

Step 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 2.3.3.2.3

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

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Step 2.3.3.2.3.1

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

$x = \sqrt{6}$

Step 2.3.3.2.3.2

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

$x = - \sqrt{6}$

Step 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 2.3.4

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

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

Step 2.4

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 points where the second derivative is $0$ .

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Step 3.1

Substitute $0$ in $f (x) = x \sqrt{4 - x^{2}}$ to find the value of $y$ .

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Step 3.1.1

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

$f (0) = (0) \sqrt{4 - {(0)}^{2}}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

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

$f (0) = 0 \sqrt{4 - 0}$

Step 3.1.2.2

Multiply $- 1$ by $0$ .

$f (0) = 0 \sqrt{4 + 0}$

Step 3.1.2.3

Add $4$ and $0$ .

$f (0) = 0 \sqrt{4}$

Step 3.1.2.4

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

$f (0) = 0 \sqrt{2^{2}}$

Step 3.1.2.5

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

$f (0) = 0 \cdot 2$

Step 3.1.2.6

Multiply $0$ by $2$ .

$f (0) = 0$

Step 3.1.2.7

The final answer is $0$ .

$0$

Step 3.2

The point found by substituting $0$ in $f (x) = x \sqrt{4 - x^{2}}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 4

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

$(- \infty, 0) \cup (0, \infty)$

Step 5

Substitute a value from the interval $(- \infty, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 0.1$ in the expression.

$f'' (- 0.1) = \frac{2 (- 0.1) ({(- 0.1)}^{2} - 6)}{{(- {(- 0.1)}^{2} + 4)}^{\frac{3}{2}}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Multiply $2$ by $- 0.1$ .

$f'' (- 0.1) = \frac{- 0.2 ({(- 0.1)}^{2} - 6)}{{(- {(- 0.1)}^{2} + 4)}^{\frac{3}{2}}}$

Step 5.2.1.2

Multiply $- 0.2$ by ${(- 0.1)}^{2} - 6$ .

$f'' (- 0.1) = \frac{1.198}{{(- {(- 0.1)}^{2} + 4)}^{\frac{3}{2}}}$

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

Raise $- 0.1$ to the power of $2$ .

$f'' (- 0.1) = \frac{1.198}{{(- 1 \cdot 0.01 + 4)}^{\frac{3}{2}}}$

Step 5.2.2.1.2

Multiply $- 1$ by $0.01$ .

$f'' (- 0.1) = \frac{1.198}{{(- 0.01 + 4)}^{\frac{3}{2}}}$

Step 5.2.2.2

Add $- 0.01$ and $4$ .

$f'' (- 0.1) = \frac{1.198}{{3.99}^{\frac{3}{2}}}$

Step 5.2.2.3

Rewrite $3.99$ as ${1.99749843}^{2}$ .

$f'' (- 0.1) = \frac{1.198}{{({1.99749843}^{2})}^{\frac{3}{2}}}$

Step 5.2.2.4

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

$f'' (- 0.1) = \frac{1.198}{{1.99749843}^{2 (\frac{3}{2})}}$

Step 5.2.2.5

Cancel the common factor of $2$ .

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Step 5.2.2.5.1

Cancel the common factor.

$f'' (- 0.1) = \frac{1.198}{{1.99749843}^{2 (\frac{3}{2})}}$

Step 5.2.2.5.2

Rewrite the expression.

$f'' (- 0.1) = \frac{1.198}{{1.99749843}^{3}}$

Step 5.2.2.6

Raise $1.99749843$ to the power of $3$ .

$f'' (- 0.1) = \frac{1.198}{7.97001875}$

Step 5.2.3

Divide $1.198$ by $7.97001875$ .

$f'' (- 0.1) = 0.15031332$

Step 5.2.4

The final answer is $0.15031332$ .

$0.15031332$

Step 5.3

At $- 0.1$ , the second derivative is $0.15031332$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, 0)$ .

Increasing on $(- \infty, 0)$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(0, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $0.1$ in the expression.

$f'' (0.1) = \frac{2 (0.1) ({(0.1)}^{2} - 6)}{{(- {(0.1)}^{2} + 4)}^{\frac{3}{2}}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Multiply $2$ by $0.1$ .

$f'' (0.1) = \frac{0.2 ({0.1}^{2} - 6)}{{(- {0.1}^{2} + 4)}^{\frac{3}{2}}}$

Step 6.2.1.2

Multiply $0.2$ by ${0.1}^{2} - 6$ .

$f'' (0.1) = \frac{- 1.198}{{(- {0.1}^{2} + 4)}^{\frac{3}{2}}}$

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

Raise $0.1$ to the power of $2$ .

$f'' (0.1) = \frac{- 1.198}{{(- 1 \cdot 0.01 + 4)}^{\frac{3}{2}}}$

Step 6.2.2.1.2

Multiply $- 1$ by $0.01$ .

$f'' (0.1) = \frac{- 1.198}{{(- 0.01 + 4)}^{\frac{3}{2}}}$

Step 6.2.2.2

Add $- 0.01$ and $4$ .

$f'' (0.1) = \frac{- 1.198}{{3.99}^{\frac{3}{2}}}$

Step 6.2.2.3

Rewrite $3.99$ as ${1.99749843}^{2}$ .

$f'' (0.1) = \frac{- 1.198}{{({1.99749843}^{2})}^{\frac{3}{2}}}$

Step 6.2.2.4

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

$f'' (0.1) = \frac{- 1.198}{{1.99749843}^{2 (\frac{3}{2})}}$

Step 6.2.2.5

Cancel the common factor of $2$ .

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Step 6.2.2.5.1

Cancel the common factor.

$f'' (0.1) = \frac{- 1.198}{{1.99749843}^{2 (\frac{3}{2})}}$

Step 6.2.2.5.2

Rewrite the expression.

$f'' (0.1) = \frac{- 1.198}{{1.99749843}^{3}}$

Step 6.2.2.6

Raise $1.99749843$ to the power of $3$ .

$f'' (0.1) = \frac{- 1.198}{7.97001875}$

Step 6.2.3

Divide $- 1.198$ by $7.97001875$ .

$f'' (0.1) = - 0.15031332$

Step 6.2.4

The final answer is $- 0.15031332$ .

$- 0.15031332$

Step 6.3

At $0.1$ , the second derivative is $- 0.15031332$ . Since this is negative, the second derivative is decreasing on the interval $(0, \infty)$

Decreasing on $(0, \infty)$ since $f'' (x) < 0$

Step 7

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 point in this case is $(0, 0)$ .

$(0, 0)$

Step 8