Calculus Examples

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

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

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

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

Move $2$ to the left of $x^{2} + 4$ .

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

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

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

Step 1.1.2.6

Simplify the expression.

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Step 1.1.2.6.1

Add $2 x$ and $0$ .

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

Step 1.1.2.6.2

Multiply $2$ by $- 1$ .

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

Add $1$ and $2$ .

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

Step 1.1.6

Simplify.

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Step 1.1.6.1

Apply the distributive property.

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

Step 1.1.6.2

Apply the distributive property.

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

Step 1.1.6.3

Simplify the numerator.

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Step 1.1.6.3.1

Simplify each term.

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Step 1.1.6.3.1.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1.6.3.1.1.1

Move $x$ .

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

Step 1.1.6.3.1.1.2

Multiply $x$ by $x^{2}$ .

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Step 1.1.6.3.1.1.2.1

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

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

Step 1.1.6.3.1.1.2.2

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

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

Step 1.1.6.3.1.1.3

Add $1$ and $2$ .

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

Step 1.1.6.3.1.2

Multiply $2$ by $4$ .

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

Step 1.1.6.3.2

Combine the opposite terms in $2 x^{3} + 8 x - 2 x^{3}$ .

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Step 1.1.6.3.2.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

$f' (x) = \frac{8 x + 0}{{(x^{2} + 4)}^{2}}$

Step 1.1.6.3.2.2

Add $8 x$ and $0$ .

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

Step 1.2

Find the second derivative.

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Step 1.2.1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8 x}{{(x^{2} + 4)}^{2}}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{x}{{(x^{2} + 4)}^{2}}]$ .

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

Step 1.2.2

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

Step 1.2.3

Differentiate using the Power Rule.

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Step 1.2.3.1

Multiply the exponents in ${({(x^{2} + 4)}^{2})}^{2}$ .

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Step 1.2.3.1.1

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

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

Step 1.2.3.1.2

Multiply $2$ by $2$ .

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

Step 1.2.3.2

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

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

Step 1.2.3.3

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

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

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

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

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

Step 1.2.4.3

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

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

Step 1.2.5

Simplify with factoring out.

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Step 1.2.5.1

Multiply $2$ by $- 1$ .

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

Step 1.2.5.2

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

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Step 1.2.5.2.1

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

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

Step 1.2.5.2.2

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

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

Step 1.2.5.2.3

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

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

Step 1.2.6

Cancel the common factors.

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Step 1.2.6.1

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

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

Step 1.2.6.2

Cancel the common factor.

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

Step 1.2.6.3

Rewrite the expression.

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

Step 1.2.7

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

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

Step 1.2.8

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

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

Step 1.2.9

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

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

Step 1.2.10

Simplify the expression.

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Step 1.2.10.1

Add $2 x$ and $0$ .

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

Step 1.2.10.2

Multiply $2$ by $- 2$ .

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

Step 1.2.11

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

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

Step 1.2.12

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

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

Step 1.2.13

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

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

Step 1.2.14

Add $1$ and $1$ .

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

Step 1.2.15

Subtract $4 x^{2}$ from $x^{2}$ .

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

Step 1.2.16

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

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

Step 1.2.17

Simplify.

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Step 1.2.17.1

Apply the distributive property.

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

Step 1.2.17.2

Simplify each term.

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Step 1.2.17.2.1

Multiply $- 3$ by $8$ .

$f'' (x) = \frac{- 24 x^{2} + 8 \cdot 4}{{(x^{2} + 4)}^{3}}$

Step 1.2.17.2.2

Multiply $8$ by $4$ .

$f'' (x) = \frac{- 24 x^{2} + 32}{{(x^{2} + 4)}^{3}}$

Step 1.2.17.3

Factor $8$ out of $- 24 x^{2} + 32$ .

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Step 1.2.17.3.1

Factor $8$ out of $- 24 x^{2}$ .

$f'' (x) = \frac{8 (- 3 x^{2}) + 32}{{(x^{2} + 4)}^{3}}$

Step 1.2.17.3.2

Factor $8$ out of $32$ .

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

Step 1.2.17.3.3

Factor $8$ out of $8 (- 3 x^{2}) + 8 (4)$ .

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

Step 1.2.17.4

Factor $- 1$ out of $- 3 x^{2}$ .

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

Step 1.2.17.5

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

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

Step 1.2.17.6

Factor $- 1$ out of $- (3 x^{2}) - 1 (- 4)$ .

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

Step 1.2.17.7

Rewrite $- (3 x^{2} - 4)$ as $- 1 (3 x^{2} - 4)$ .

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

Step 1.2.17.8

Move the negative in front of the fraction.

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

Step 1.3

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

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

Step 2

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

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Step 2.1

Set the second derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

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

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Divide each term in $8 (3 x^{2} - 4) = 0$ by $8$ and simplify.

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Step 2.3.1.1

Divide each term in $8 (3 x^{2} - 4) = 0$ by $8$ .

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

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $8$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

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

Step 2.3.1.2.1.2

Divide $3 x^{2} - 4$ by $1$ .

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

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $8$ .

$3 x^{2} - 4 = 0$

Step 2.3.2

Add $4$ to both sides of the equation.

$3 x^{2} = 4$

Step 2.3.3

Divide each term in $3 x^{2} = 4$ by $3$ and simplify.

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Step 2.3.3.1

Divide each term in $3 x^{2} = 4$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{4}{3}$

Step 2.3.3.2

Simplify the left side.

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Step 2.3.3.2.1

Cancel the common factor of $3$ .

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Step 2.3.3.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{4}{3}$

Step 2.3.3.2.1.2

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

$x^{2} = \frac{4}{3}$

Step 2.3.4

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

$x = \pm \sqrt{\frac{4}{3}}$

Step 2.3.5

Simplify $\pm \sqrt{\frac{4}{3}}$ .

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Step 2.3.5.1

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{4}}{\sqrt{3}}$

Step 2.3.5.2

Simplify the numerator.

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Step 2.3.5.2.1

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

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

Step 2.3.5.2.2

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

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

Step 2.3.5.3

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

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

Step 2.3.5.4

Combine and simplify the denominator.

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Step 2.3.5.4.1

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

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

Step 2.3.5.4.2

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

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

Step 2.3.5.4.3

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

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

Step 2.3.5.4.4

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

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

Step 2.3.5.4.5

Add $1$ and $1$ .

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

Step 2.3.5.4.6

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

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Step 2.3.5.4.6.1

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

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

Step 2.3.5.4.6.2

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

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

Step 2.3.5.4.6.3

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

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

Step 2.3.5.4.6.4

Cancel the common factor of $2$ .

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Step 2.3.5.4.6.4.1

Cancel the common factor.

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

Step 2.3.5.4.6.4.2

Rewrite the expression.

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

Step 2.3.5.4.6.5

Evaluate the exponent.

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

Step 2.3.6

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

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Step 2.3.6.1

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

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

Step 2.3.6.2

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

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

Step 2.3.6.3

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

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

Step 3

Find the points where the second derivative is $0$ .

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Step 3.1

Substitute $\frac{2 \sqrt{3}}{3}$ in $f (x) = \frac{x^{2}}{x^{2} + 4}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $\frac{2 \sqrt{3}}{3}$ in the expression.

$f (\frac{2 \sqrt{3}}{3}) = \frac{{(\frac{2 \sqrt{3}}{3})}^{2}}{{(\frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify the numerator.

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Step 3.1.2.1.1

Apply the product rule to $\frac{2 \sqrt{3}}{3}$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{{(2 \sqrt{3})}^{2}}{3^{2}}}{{(\frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.1.2.1.2

Simplify the numerator.

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Step 3.1.2.1.2.1

Apply the product rule to $2 \sqrt{3}$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{2^{2} {\sqrt{3}}^{2}}{3^{2}}}{{(\frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.1.2.1.2.2

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

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4 {\sqrt{3}}^{2}}{3^{2}}}{{(\frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.1.2.1.2.3

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

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Step 3.1.2.1.2.3.1

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

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4 {(3^{\frac{1}{2}})}^{2}}{3^{2}}}{{(\frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.1.2.1.2.3.2

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

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

Step 3.1.2.1.2.3.3

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

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4 \cdot 3^{\frac{2}{2}}}{3^{2}}}{{(\frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.1.2.1.2.3.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.2.3.4.1

Cancel the common factor.

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4 \cdot 3^{\frac{2}{2}}}{3^{2}}}{{(\frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.1.2.1.2.3.4.2

Rewrite the expression.

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4 \cdot 3}{3^{2}}}{{(\frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.1.2.1.2.3.5

Evaluate the exponent.

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4 \cdot 3}{3^{2}}}{{(\frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.1.2.1.3

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

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4 \cdot 3}{9}}{{(\frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.1.2.1.4

Multiply $4$ by $3$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{12}{9}}{{(\frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.1.2.1.5

Cancel the common factor of $12$ and $9$ .

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Step 3.1.2.1.5.1

Factor $3$ out of $12$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{3 (4)}{9}}{{(\frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.1.2.1.5.2

Cancel the common factors.

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Step 3.1.2.1.5.2.1

Factor $3$ out of $9$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{3 \cdot 4}{3 \cdot 3}}{{(\frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.1.2.1.5.2.2

Cancel the common factor.

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{3 \cdot 4}{3 \cdot 3}}{{(\frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.1.2.1.5.2.3

Rewrite the expression.

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{{(\frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.1.2.2

Simplify the denominator.

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Step 3.1.2.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.2.2.1.1

Apply the product rule to $\frac{2 \sqrt{3}}{3}$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{{(2 \sqrt{3})}^{2}}{3^{2}} + 4}$

Step 3.1.2.2.1.2

Apply the product rule to $2 \sqrt{3}$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{2^{2} {\sqrt{3}}^{2}}{3^{2}} + 4}$

Step 3.1.2.2.2

Simplify the numerator.

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Step 3.1.2.2.2.1

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

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4 {\sqrt{3}}^{2}}{3^{2}} + 4}$

Step 3.1.2.2.2.2

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

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Step 3.1.2.2.2.2.1

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

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4 {(3^{\frac{1}{2}})}^{2}}{3^{2}} + 4}$

Step 3.1.2.2.2.2.2

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

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

Step 3.1.2.2.2.2.3

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

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4 \cdot 3^{\frac{2}{2}}}{3^{2}} + 4}$

Step 3.1.2.2.2.2.4

Cancel the common factor of $2$ .

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Step 3.1.2.2.2.2.4.1

Cancel the common factor.

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4 \cdot 3^{\frac{2}{2}}}{3^{2}} + 4}$

Step 3.1.2.2.2.2.4.2

Rewrite the expression.

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4 \cdot 3}{3^{2}} + 4}$

Step 3.1.2.2.2.2.5

Evaluate the exponent.

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4 \cdot 3}{3^{2}} + 4}$

Step 3.1.2.2.3

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

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4 \cdot 3}{9} + 4}$

Step 3.1.2.2.4

Multiply $4$ by $3$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{12}{9} + 4}$

Step 3.1.2.2.5

Cancel the common factor of $12$ and $9$ .

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Step 3.1.2.2.5.1

Factor $3$ out of $12$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{3 (4)}{9} + 4}$

Step 3.1.2.2.5.2

Cancel the common factors.

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Step 3.1.2.2.5.2.1

Factor $3$ out of $9$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{3 \cdot 4}{3 \cdot 3} + 4}$

Step 3.1.2.2.5.2.2

Cancel the common factor.

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{3 \cdot 4}{3 \cdot 3} + 4}$

Step 3.1.2.2.5.2.3

Rewrite the expression.

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4}{3} + 4}$

Step 3.1.2.2.6

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4}{3} + 4 \cdot \frac{3}{3}}$

Step 3.1.2.2.7

Combine $4$ and $\frac{3}{3}$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4}{3} + \frac{4 \cdot 3}{3}}$

Step 3.1.2.2.8

Combine the numerators over the common denominator.

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4 + 4 \cdot 3}{3}}$

Step 3.1.2.2.9

Simplify the numerator.

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Step 3.1.2.2.9.1

Multiply $4$ by $3$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4 + 12}{3}}$

Step 3.1.2.2.9.2

Add $4$ and $12$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{16}{3}}$

Step 3.1.2.3

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{2 \sqrt{3}}{3}) = \frac{4}{3} \cdot \frac{3}{16}$

Step 3.1.2.4

Cancel the common factor of $4$ .

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Step 3.1.2.4.1

Factor $4$ out of $16$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{4}{3} \cdot \frac{3}{4 (4)}$

Step 3.1.2.4.2

Cancel the common factor.

$f (\frac{2 \sqrt{3}}{3}) = \frac{4}{3} \cdot \frac{3}{4 \cdot 4}$

Step 3.1.2.4.3

Rewrite the expression.

$f (\frac{2 \sqrt{3}}{3}) = \frac{1}{3} \cdot \frac{3}{4}$

Step 3.1.2.5

Cancel the common factor of $3$ .

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Step 3.1.2.5.1

Cancel the common factor.

$f (\frac{2 \sqrt{3}}{3}) = \frac{1}{3} \cdot \frac{3}{4}$

Step 3.1.2.5.2

Rewrite the expression.

$f (\frac{2 \sqrt{3}}{3}) = \frac{1}{4}$

Step 3.1.2.6

The final answer is $\frac{1}{4}$ .

$\frac{1}{4}$

Step 3.2

The point found by substituting $\frac{2 \sqrt{3}}{3}$ in $f (x) = \frac{x^{2}}{x^{2} + 4}$ is $(\frac{2 \sqrt{3}}{3}, \frac{1}{4})$ . This point can be an inflection point.

$(\frac{2 \sqrt{3}}{3}, \frac{1}{4})$

Step 3.3

Substitute $- \frac{2 \sqrt{3}}{3}$ in $f (x) = \frac{x^{2}}{x^{2} + 4}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $- \frac{2 \sqrt{3}}{3}$ in the expression.

$f (- \frac{2 \sqrt{3}}{3}) = \frac{{(- \frac{2 \sqrt{3}}{3})}^{2}}{{(- \frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify the numerator.

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Step 3.3.2.1.1

Apply the product rule to $- \frac{2 \sqrt{3}}{3}$ .

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

Step 3.3.2.1.2

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

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

Step 3.3.2.1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.3.2.1.3.1

Apply the product rule to $\frac{2 \sqrt{3}}{3}$ .

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

Step 3.3.2.1.3.2

Apply the product rule to $2 \sqrt{3}$ .

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

Step 3.3.2.1.4

Simplify the numerator.

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Step 3.3.2.1.4.1

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

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

Step 3.3.2.1.4.2

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

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Step 3.3.2.1.4.2.1

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

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

Step 3.3.2.1.4.2.2

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

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

Step 3.3.2.1.4.2.3

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

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

Step 3.3.2.1.4.2.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.4.2.4.1

Cancel the common factor.

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

Step 3.3.2.1.4.2.4.2

Rewrite the expression.

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

Step 3.3.2.1.4.2.5

Evaluate the exponent.

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

Step 3.3.2.1.5

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

$f (- \frac{2 \sqrt{3}}{3}) = \frac{1 (\frac{4 \cdot 3}{9})}{{(- \frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.3.2.1.6

Multiply $4$ by $3$ .

$f (- \frac{2 \sqrt{3}}{3}) = \frac{1 (\frac{12}{9})}{{(- \frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.3.2.1.7

Cancel the common factor of $12$ and $9$ .

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Step 3.3.2.1.7.1

Factor $3$ out of $12$ .

$f (- \frac{2 \sqrt{3}}{3}) = \frac{1 (\frac{3 (4)}{9})}{{(- \frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.3.2.1.7.2

Cancel the common factors.

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Step 3.3.2.1.7.2.1

Factor $3$ out of $9$ .

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

Step 3.3.2.1.7.2.2

Cancel the common factor.

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

Step 3.3.2.1.7.2.3

Rewrite the expression.

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

Step 3.3.2.1.8

Multiply $\frac{4}{3}$ by $1$ .

$f (- \frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{{(- \frac{2 \sqrt{3}}{3})}^{2} + 4}$

Step 3.3.2.2

Simplify the denominator.

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Step 3.3.2.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.3.2.2.1.1

Apply the product rule to $- \frac{2 \sqrt{3}}{3}$ .

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

Step 3.3.2.2.1.2

Apply the product rule to $\frac{2 \sqrt{3}}{3}$ .

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

Step 3.3.2.2.1.3

Apply the product rule to $2 \sqrt{3}$ .

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

Step 3.3.2.2.2

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

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

Step 3.3.2.2.3

Multiply $\frac{2^{2} {\sqrt{3}}^{2}}{3^{2}}$ by $1$ .

$f (- \frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{2^{2} {\sqrt{3}}^{2}}{3^{2}} + 4}$

Step 3.3.2.2.4

Simplify the numerator.

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Step 3.3.2.2.4.1

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

$f (- \frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4 {\sqrt{3}}^{2}}{3^{2}} + 4}$

Step 3.3.2.2.4.2

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

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Step 3.3.2.2.4.2.1

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

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

Step 3.3.2.2.4.2.2

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

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

Step 3.3.2.2.4.2.3

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

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

Step 3.3.2.2.4.2.4

Cancel the common factor of $2$ .

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Step 3.3.2.2.4.2.4.1

Cancel the common factor.

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

Step 3.3.2.2.4.2.4.2

Rewrite the expression.

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

Step 3.3.2.2.4.2.5

Evaluate the exponent.

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

Step 3.3.2.2.5

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

$f (- \frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4 \cdot 3}{9} + 4}$

Step 3.3.2.2.6

Multiply $4$ by $3$ .

$f (- \frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{12}{9} + 4}$

Step 3.3.2.2.7

Cancel the common factor of $12$ and $9$ .

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Step 3.3.2.2.7.1

Factor $3$ out of $12$ .

$f (- \frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{3 (4)}{9} + 4}$

Step 3.3.2.2.7.2

Cancel the common factors.

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Step 3.3.2.2.7.2.1

Factor $3$ out of $9$ .

$f (- \frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{3 \cdot 4}{3 \cdot 3} + 4}$

Step 3.3.2.2.7.2.2

Cancel the common factor.

$f (- \frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{3 \cdot 4}{3 \cdot 3} + 4}$

Step 3.3.2.2.7.2.3

Rewrite the expression.

$f (- \frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4}{3} + 4}$

Step 3.3.2.2.8

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- \frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4}{3} + 4 \cdot \frac{3}{3}}$

Step 3.3.2.2.9

Combine $4$ and $\frac{3}{3}$ .

$f (- \frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4}{3} + \frac{4 \cdot 3}{3}}$

Step 3.3.2.2.10

Combine the numerators over the common denominator.

$f (- \frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4 + 4 \cdot 3}{3}}$

Step 3.3.2.2.11

Simplify the numerator.

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Step 3.3.2.2.11.1

Multiply $4$ by $3$ .

$f (- \frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{4 + 12}{3}}$

Step 3.3.2.2.11.2

Add $4$ and $12$ .

$f (- \frac{2 \sqrt{3}}{3}) = \frac{\frac{4}{3}}{\frac{16}{3}}$

Step 3.3.2.3

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{2 \sqrt{3}}{3}) = \frac{4}{3} \cdot \frac{3}{16}$

Step 3.3.2.4

Cancel the common factor of $4$ .

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Step 3.3.2.4.1

Factor $4$ out of $16$ .

$f (- \frac{2 \sqrt{3}}{3}) = \frac{4}{3} \cdot \frac{3}{4 (4)}$

Step 3.3.2.4.2

Cancel the common factor.

$f (- \frac{2 \sqrt{3}}{3}) = \frac{4}{3} \cdot \frac{3}{4 \cdot 4}$

Step 3.3.2.4.3

Rewrite the expression.

$f (- \frac{2 \sqrt{3}}{3}) = \frac{1}{3} \cdot \frac{3}{4}$

Step 3.3.2.5

Cancel the common factor of $3$ .

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Step 3.3.2.5.1

Cancel the common factor.

$f (- \frac{2 \sqrt{3}}{3}) = \frac{1}{3} \cdot \frac{3}{4}$

Step 3.3.2.5.2

Rewrite the expression.

$f (- \frac{2 \sqrt{3}}{3}) = \frac{1}{4}$

Step 3.3.2.6

The final answer is $\frac{1}{4}$ .

$\frac{1}{4}$

Step 3.4

The point found by substituting $- \frac{2 \sqrt{3}}{3}$ in $f (x) = \frac{x^{2}}{x^{2} + 4}$ is $(- \frac{2 \sqrt{3}}{3}, \frac{1}{4})$ . This point can be an inflection point.

$(- \frac{2 \sqrt{3}}{3}, \frac{1}{4})$

Step 3.5

Determine the points that could be inflection points.

$(\frac{2 \sqrt{3}}{3}, \frac{1}{4}), (- \frac{2 \sqrt{3}}{3}, \frac{1}{4})$

Step 4

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

$(- \infty, - \frac{2 \sqrt{3}}{3}) \cup (- \frac{2 \sqrt{3}}{3}, \frac{2 \sqrt{3}}{3}) \cup (\frac{2 \sqrt{3}}{3}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - \frac{2 \sqrt{3}}{3})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 1.25470053$ in the expression.

$f'' (- 1.25470053) = - \frac{8 (3 {(- 1.25470053)}^{2} - 4)}{{({(- 1.25470053)}^{2} + 4)}^{3}}$

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

Raise $- 1.25470053$ to the power of $2$ .

$f'' (- 1.25470053) = - \frac{8 (3 \cdot 1.57427344 - 4)}{{({(- 1.25470053)}^{2} + 4)}^{3}}$

Step 5.2.1.2

Multiply $3$ by $1.57427344$ .

$f'' (- 1.25470053) = - \frac{8 (4.72282032 - 4)}{{({(- 1.25470053)}^{2} + 4)}^{3}}$

Step 5.2.1.3

Subtract $4$ from $4.72282032$ .

$f'' (- 1.25470053) = - \frac{8 \cdot 0.72282032}{{({(- 1.25470053)}^{2} + 4)}^{3}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Raise $- 1.25470053$ to the power of $2$ .

$f'' (- 1.25470053) = - \frac{8 \cdot 0.72282032}{{(1.57427344 + 4)}^{3}}$

Step 5.2.2.2

Add $1.57427344$ and $4$ .

$f'' (- 1.25470053) = - \frac{8 \cdot 0.72282032}{{5.57427344}^{3}}$

Step 5.2.2.3

Raise $5.57427344$ to the power of $3$ .

$f'' (- 1.25470053) = - \frac{8 \cdot 0.72282032}{173.20674748}$

Step 5.2.3

Simplify the expression.

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Step 5.2.3.1

Multiply $8$ by $0.72282032$ .

$f'' (- 1.25470053) = - \frac{5.78256258}{173.20674748}$

Step 5.2.3.2

Divide $5.78256258$ by $173.20674748$ .

$f'' (- 1.25470053) = - 1 \cdot 0.03338531$

Step 5.2.3.3

Multiply $- 1$ by $0.03338531$ .

$f'' (- 1.25470053) = - 0.03338531$

Step 5.2.4

The final answer is $- 0.03338531$ .

$- 0.03338531$

Step 5.3

At $- 1.25470053$ , the second derivative is $- 0.03338531$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - \frac{2 \sqrt{3}}{3})$

Decreasing on $(- \infty, - \frac{2 \sqrt{3}}{3})$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(- \frac{2 \sqrt{3}}{3}, \frac{2 \sqrt{3}}{3})$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

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

$f'' (0) = - \frac{8 (3 {(0)}^{2} - 4)}{{({(0)}^{2} + 4)}^{3}}$

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

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

$f'' (0) = - \frac{8 (3 \cdot 0 - 4)}{{(0^{2} + 4)}^{3}}$

Step 6.2.1.2

Multiply $3$ by $0$ .

$f'' (0) = - \frac{8 (0 - 4)}{{(0^{2} + 4)}^{3}}$

Step 6.2.1.3

Subtract $4$ from $0$ .

$f'' (0) = - \frac{8 \cdot - 4}{{(0^{2} + 4)}^{3}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

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

$f'' (0) = - \frac{8 \cdot - 4}{{(0 + 4)}^{3}}$

Step 6.2.2.2

Add $0$ and $4$ .

$f'' (0) = - \frac{8 \cdot - 4}{4^{3}}$

Step 6.2.2.3

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

$f'' (0) = - \frac{8 \cdot - 4}{64}$

Step 6.2.3

Reduce the expression by cancelling the common factors.

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Step 6.2.3.1

Multiply $8$ by $- 4$ .

$f'' (0) = - \frac{- 32}{64}$

Step 6.2.3.2

Cancel the common factor of $- 32$ and $64$ .

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Step 6.2.3.2.1

Factor $32$ out of $- 32$ .

$f'' (0) = - \frac{32 (- 1)}{64}$

Step 6.2.3.2.2

Cancel the common factors.

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Step 6.2.3.2.2.1

Factor $32$ out of $64$ .

$f'' (0) = - \frac{32 \cdot - 1}{32 \cdot 2}$

Step 6.2.3.2.2.2

Cancel the common factor.

$f'' (0) = - \frac{32 \cdot - 1}{32 \cdot 2}$

Step 6.2.3.2.2.3

Rewrite the expression.

$f'' (0) = - \frac{- 1}{2}$

Step 6.2.3.3

Move the negative in front of the fraction.

$f'' (0) = \frac{1}{2}$

Step 6.2.4

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

Step 6.3

At $0$ , the second derivative is $0.5$ . Since this is positive, the second derivative is increasing on the interval $(- \frac{2 \sqrt{3}}{3}, \frac{2 \sqrt{3}}{3})$ .

Increasing on $(- \frac{2 \sqrt{3}}{3}, \frac{2 \sqrt{3}}{3})$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(\frac{2 \sqrt{3}}{3}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $1.25470053$ in the expression.

$f'' (1.25470053) = - \frac{8 (3 {(1.25470053)}^{2} - 4)}{{({(1.25470053)}^{2} + 4)}^{3}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Raise $1.25470053$ to the power of $2$ .

$f'' (1.25470053) = - \frac{8 (3 \cdot 1.57427344 - 4)}{{({1.25470053}^{2} + 4)}^{3}}$

Step 7.2.1.2

Multiply $3$ by $1.57427344$ .

$f'' (1.25470053) = - \frac{8 (4.72282032 - 4)}{{({1.25470053}^{2} + 4)}^{3}}$

Step 7.2.1.3

Subtract $4$ from $4.72282032$ .

$f'' (1.25470053) = - \frac{8 \cdot 0.72282032}{{({1.25470053}^{2} + 4)}^{3}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Raise $1.25470053$ to the power of $2$ .

$f'' (1.25470053) = - \frac{8 \cdot 0.72282032}{{(1.57427344 + 4)}^{3}}$

Step 7.2.2.2

Add $1.57427344$ and $4$ .

$f'' (1.25470053) = - \frac{8 \cdot 0.72282032}{{5.57427344}^{3}}$

Step 7.2.2.3

Raise $5.57427344$ to the power of $3$ .

$f'' (1.25470053) = - \frac{8 \cdot 0.72282032}{173.20674748}$

Step 7.2.3

Simplify the expression.

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Step 7.2.3.1

Multiply $8$ by $0.72282032$ .

$f'' (1.25470053) = - \frac{5.78256258}{173.20674748}$

Step 7.2.3.2

Divide $5.78256258$ by $173.20674748$ .

$f'' (1.25470053) = - 1 \cdot 0.03338531$

Step 7.2.3.3

Multiply $- 1$ by $0.03338531$ .

$f'' (1.25470053) = - 0.03338531$

Step 7.2.4

The final answer is $- 0.03338531$ .

$- 0.03338531$

Step 7.3

At $1.25470053$ , the second derivative is $- 0.03338531$ . Since this is negative, the second derivative is decreasing on the interval $(\frac{2 \sqrt{3}}{3}, \infty)$

Decreasing on $(\frac{2 \sqrt{3}}{3}, \infty)$ since $f'' (x) < 0$

Step 8

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 points in this case are $(- \frac{2 \sqrt{3}}{3}, \frac{1}{4}), (\frac{2 \sqrt{3}}{3}, \frac{1}{4})$ .

$(- \frac{2 \sqrt{3}}{3}, \frac{1}{4}), (\frac{2 \sqrt{3}}{3}, \frac{1}{4})$

Step 9