Calculus Examples

$\frac{x^{2}}{x^{2} + 9}$

Step 1

Write $\frac{x^{2}}{x^{2} + 9}$ as a function.

$f (x) = \frac{x^{2}}{x^{2} + 9}$

Step 2

Find the first derivative of the function.

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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 $g (x) = x^{2} + 9$ .

$\frac{(x^{2} + 9) \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [x^{2} + 9]}{{(x^{2} + 9)}^{2}}$

Differentiate.

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

$\frac{(x^{2} + 9) (2 x) - x^{2} \frac{d}{d x} [x^{2} + 9]}{{(x^{2} + 9)}^{2}}$

Move $2$ to the left of $x^{2} + 9$ .

$\frac{2 \cdot (x^{2} + 9) x - x^{2} \frac{d}{d x} [x^{2} + 9]}{{(x^{2} + 9)}^{2}}$

By the Sum Rule, the derivative of $x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$ .

$\frac{2 (x^{2} + 9) x - x^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9])}{{(x^{2} + 9)}^{2}}$

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

$\frac{2 (x^{2} + 9) x - x^{2} (2 x + \frac{d}{d x} [9])}{{(x^{2} + 9)}^{2}}$

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{2 (x^{2} + 9) x - x^{2} (2 x + 0)}{{(x^{2} + 9)}^{2}}$

Simplify the expression.

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Add $2 x$ and $0$ .

$\frac{2 (x^{2} + 9) x - x^{2} (2 x)}{{(x^{2} + 9)}^{2}}$

Multiply $2$ by $- 1$ .

$\frac{2 (x^{2} + 9) x - 2 x^{2} x}{{(x^{2} + 9)}^{2}}$

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

$\frac{2 (x^{2} + 9) x - 2 (x^{1} x^{2})}{{(x^{2} + 9)}^{2}}$

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

$\frac{2 (x^{2} + 9) x - 2 x^{1 + 2}}{{(x^{2} + 9)}^{2}}$

Add $1$ and $2$ .

$\frac{2 (x^{2} + 9) x - 2 x^{3}}{{(x^{2} + 9)}^{2}}$

Simplify.

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Apply the distributive property.

$\frac{(2 x^{2} + 2 \cdot 9) x - 2 x^{3}}{{(x^{2} + 9)}^{2}}$

Apply the distributive property.

$\frac{2 x^{2} x + 2 \cdot 9 x - 2 x^{3}}{{(x^{2} + 9)}^{2}}$

Simplify the numerator.

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Simplify each term.

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

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Move $x$ .

$\frac{2 (x \cdot x^{2}) + 2 \cdot 9 x - 2 x^{3}}{{(x^{2} + 9)}^{2}}$

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

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

$\frac{2 (x^{1} x^{2}) + 2 \cdot 9 x - 2 x^{3}}{{(x^{2} + 9)}^{2}}$

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

$\frac{2 x^{1 + 2} + 2 \cdot 9 x - 2 x^{3}}{{(x^{2} + 9)}^{2}}$

Add $1$ and $2$ .

$\frac{2 x^{3} + 2 \cdot 9 x - 2 x^{3}}{{(x^{2} + 9)}^{2}}$

Multiply $2$ by $9$ .

$\frac{2 x^{3} + 18 x - 2 x^{3}}{{(x^{2} + 9)}^{2}}$

Combine the opposite terms in $2 x^{3} + 18 x - 2 x^{3}$ .

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

$\frac{18 x + 0}{{(x^{2} + 9)}^{2}}$

Add $18 x$ and $0$ .

$\frac{18 x}{{(x^{2} + 9)}^{2}}$

Step 3

Find the second derivative of the function.

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Since $18$ is constant with respect to $x$ , the derivative of $\frac{18 x}{{(x^{2} + 9)}^{2}}$ with respect to $x$ is $18 \frac{d}{d x} [\frac{x}{{(x^{2} + 9)}^{2}}]$ .

$f'' (x) = 18 \frac{d}{d x} (\frac{x}{{(x^{2} + 9)}^{2}})$

$f'' (x) = 18 (\frac{{(x^{2} + 9)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} + 9)}^{2}}{{({(x^{2} + 9)}^{2})}^{2}})$

Differentiate using the Power Rule.

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Multiply the exponents in ${({(x^{2} + 9)}^{2})}^{2}$ .

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

$f'' (x) = 18 (\frac{{(x^{2} + 9)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} + 9)}^{2}}{{(x^{2} + 9)}^{2 \cdot 2}})$

Multiply $2$ by $2$ .

$f'' (x) = 18 (\frac{{(x^{2} + 9)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} + 9)}^{2}}{{(x^{2} + 9)}^{4}})$

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

$f'' (x) = 18 (\frac{{(x^{2} + 9)}^{2} \cdot 1 - x \frac{d}{d x} {(x^{2} + 9)}^{2}}{{(x^{2} + 9)}^{4}})$

Multiply ${(x^{2} + 9)}^{2}$ by $1$ .

$f'' (x) = 18 (\frac{{(x^{2} + 9)}^{2} - x \frac{d}{d x} {(x^{2} + 9)}^{2}}{{(x^{2} + 9)}^{4}})$

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 $g (x) = x^{2} + 9$ .

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To apply the Chain Rule, set $u$ as $x^{2} + 9$ .

$f'' (x) = 18 (\frac{{(x^{2} + 9)}^{2} - x (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{2} + 9))}{{(x^{2} + 9)}^{4}})$

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

$f'' (x) = 18 (\frac{{(x^{2} + 9)}^{2} - x (2 u \frac{d}{d x} (x^{2} + 9))}{{(x^{2} + 9)}^{4}})$

Replace all occurrences of $u$ with $x^{2} + 9$ .

$f'' (x) = 18 (\frac{{(x^{2} + 9)}^{2} - x (2 (x^{2} + 9) \frac{d}{d x} (x^{2} + 9))}{{(x^{2} + 9)}^{4}})$

Simplify with factoring out.

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Multiply $2$ by $- 1$ .

$f'' (x) = 18 (\frac{{(x^{2} + 9)}^{2} - 2 x ((x^{2} + 9) \frac{d}{d x} (x^{2} + 9))}{{(x^{2} + 9)}^{4}})$

Factor $x^{2} + 9$ out of ${(x^{2} + 9)}^{2} - 2 x ((x^{2} + 9) \frac{d}{d x} [x^{2} + 9])$ .

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Factor $x^{2} + 9$ out of ${(x^{2} + 9)}^{2}$ .

$f'' (x) = 18 (\frac{(x^{2} + 9) (x^{2} + 9) - 2 x ((x^{2} + 9) \frac{d}{d x} (x^{2} + 9))}{{(x^{2} + 9)}^{4}})$

Factor $x^{2} + 9$ out of $- 2 x ((x^{2} + 9) \frac{d}{d x} [x^{2} + 9])$ .

$f'' (x) = 18 (\frac{(x^{2} + 9) (x^{2} + 9) + (x^{2} + 9) (- 2 x (\frac{d}{d x} (x^{2} + 9)))}{{(x^{2} + 9)}^{4}})$

Factor $x^{2} + 9$ out of $(x^{2} + 9) (x^{2} + 9) + (x^{2} + 9) (- 2 x (\frac{d}{d x} [x^{2} + 9]))$ .

$f'' (x) = 18 (\frac{(x^{2} + 9) (x^{2} + 9 - 2 x (\frac{d}{d x} (x^{2} + 9)))}{{(x^{2} + 9)}^{4}})$

Cancel the common factors.

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Factor $x^{2} + 9$ out of ${(x^{2} + 9)}^{4}$ .

$f'' (x) = 18 (\frac{(x^{2} + 9) (x^{2} + 9 - 2 x (\frac{d}{d x} (x^{2} + 9)))}{(x^{2} + 9) {(x^{2} + 9)}^{3}})$

Cancel the common factor.

$f'' (x) = 18 (\frac{(x^{2} + 9) (x^{2} + 9 - 2 x (\frac{d}{d x} (x^{2} + 9)))}{(x^{2} + 9) {(x^{2} + 9)}^{3}})$

Rewrite the expression.

$f'' (x) = 18 (\frac{x^{2} + 9 - 2 x (\frac{d}{d x} (x^{2} + 9))}{{(x^{2} + 9)}^{3}})$

By the Sum Rule, the derivative of $x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$ .

$f'' (x) = 18 (\frac{x^{2} + 9 - 2 x (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (9))}{{(x^{2} + 9)}^{3}})$

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

$f'' (x) = 18 (\frac{x^{2} + 9 - 2 x (2 x + \frac{d}{d x} (9))}{{(x^{2} + 9)}^{3}})$

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$f'' (x) = 18 (\frac{x^{2} + 9 - 2 x (2 x + 0)}{{(x^{2} + 9)}^{3}})$

Simplify the expression.

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Add $2 x$ and $0$ .

$f'' (x) = 18 (\frac{x^{2} + 9 - 2 x (2 x)}{{(x^{2} + 9)}^{3}})$

Multiply $2$ by $- 2$ .

$f'' (x) = 18 (\frac{x^{2} + 9 - 4 x \cdot x}{{(x^{2} + 9)}^{3}})$

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

$f'' (x) = 18 (\frac{x^{2} + 9 - 4 (x \cdot x)}{{(x^{2} + 9)}^{3}})$

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

$f'' (x) = 18 (\frac{x^{2} + 9 - 4 (x \cdot x)}{{(x^{2} + 9)}^{3}})$

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

$f'' (x) = 18 (\frac{x^{2} + 9 - 4 x^{1 + 1}}{{(x^{2} + 9)}^{3}})$

Add $1$ and $1$ .

$f'' (x) = 18 (\frac{x^{2} + 9 - 4 x^{2}}{{(x^{2} + 9)}^{3}})$

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

$f'' (x) = 18 (\frac{- 3 x^{2} + 9}{{(x^{2} + 9)}^{3}})$

Combine $18$ and $\frac{- 3 x^{2} + 9}{{(x^{2} + 9)}^{3}}$ .

$f'' (x) = \frac{18 (- 3 x^{2} + 9)}{{(x^{2} + 9)}^{3}}$

Simplify.

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Apply the distributive property.

$f'' (x) = \frac{18 (- 3 x^{2}) + 18 \cdot 9}{{(x^{2} + 9)}^{3}}$

Simplify each term.

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Multiply $- 3$ by $18$ .

$f'' (x) = \frac{- 54 x^{2} + 18 \cdot 9}{{(x^{2} + 9)}^{3}}$

Multiply $18$ by $9$ .

$f'' (x) = \frac{- 54 x^{2} + 162}{{(x^{2} + 9)}^{3}}$

Factor $54$ out of $- 54 x^{2} + 162$ .

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Factor $54$ out of $- 54 x^{2}$ .

$f'' (x) = \frac{54 (- x^{2}) + 162}{{(x^{2} + 9)}^{3}}$

Factor $54$ out of $162$ .

$f'' (x) = \frac{54 (- x^{2}) + 54 (3)}{{(x^{2} + 9)}^{3}}$

Factor $54$ out of $54 (- x^{2}) + 54 (3)$ .

$f'' (x) = \frac{54 (- x^{2} + 3)}{{(x^{2} + 9)}^{3}}$

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

$f'' (x) = \frac{54 (- (x^{2}) + 3)}{{(x^{2} + 9)}^{3}}$

Rewrite $3$ as $- 1 (- 3)$ .

$f'' (x) = \frac{54 (- (x^{2}) - 1 \cdot - 3)}{{(x^{2} + 9)}^{3}}$

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

$f'' (x) = \frac{54 (- (x^{2} - 3))}{{(x^{2} + 9)}^{3}}$

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

$f'' (x) = \frac{54 (- 1 (x^{2} - 3))}{{(x^{2} + 9)}^{3}}$

Move the negative in front of the fraction.

$f'' (x) = - \frac{54 (x^{2} - 3)}{{(x^{2} + 9)}^{3}}$

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{18 x}{{(x^{2} + 9)}^{2}} = 0$

Step 5

Find the first derivative.

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Find the first derivative.

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$f' (x) = \frac{(x^{2} + 9) \frac{d}{d x} (x^{2}) - x^{2} \frac{d}{d x} (x^{2} + 9)}{{(x^{2} + 9)}^{2}}$

Differentiate.

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

$f' (x) = \frac{(x^{2} + 9) (2 x) - x^{2} \frac{d}{d x} (x^{2} + 9)}{{(x^{2} + 9)}^{2}}$

Move $2$ to the left of $x^{2} + 9$ .

$f' (x) = \frac{2 \cdot ((x^{2} + 9) x) - x^{2} \frac{d}{d x} (x^{2} + 9)}{{(x^{2} + 9)}^{2}}$

By the Sum Rule, the derivative of $x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$ .

$f' (x) = \frac{2 (x^{2} + 9) x - x^{2} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (9))}{{(x^{2} + 9)}^{2}}$

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

$f' (x) = \frac{2 (x^{2} + 9) x - x^{2} (2 x + \frac{d}{d x} (9))}{{(x^{2} + 9)}^{2}}$

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$f' (x) = \frac{2 (x^{2} + 9) x - x^{2} (2 x + 0)}{{(x^{2} + 9)}^{2}}$

Simplify the expression.

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Add $2 x$ and $0$ .

$f' (x) = \frac{2 (x^{2} + 9) x - x^{2} (2 x)}{{(x^{2} + 9)}^{2}}$

Multiply $2$ by $- 1$ .

$f' (x) = \frac{2 (x^{2} + 9) x - 2 x^{2} x}{{(x^{2} + 9)}^{2}}$

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

$f' (x) = \frac{2 (x^{2} + 9) x - 2 (x \cdot x^{2})}{{(x^{2} + 9)}^{2}}$

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

$f' (x) = \frac{2 (x^{2} + 9) x - 2 x^{1 + 2}}{{(x^{2} + 9)}^{2}}$

Add $1$ and $2$ .

$f' (x) = \frac{2 (x^{2} + 9) x - 2 x^{3}}{{(x^{2} + 9)}^{2}}$

Simplify.

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Apply the distributive property.

$f' (x) = \frac{(2 x^{2} + 2 \cdot 9) x - 2 x^{3}}{{(x^{2} + 9)}^{2}}$

Apply the distributive property.

$f' (x) = \frac{2 x^{2} x + 2 \cdot (9 x) - 2 x^{3}}{{(x^{2} + 9)}^{2}}$

Simplify the numerator.

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Simplify each term.

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

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Move $x$ .

$f' (x) = \frac{2 (x \cdot x^{2}) + 2 \cdot (9 x) - 2 x^{3}}{{(x^{2} + 9)}^{2}}$

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

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

$f' (x) = \frac{2 (x \cdot x^{2}) + 2 \cdot (9 x) - 2 x^{3}}{{(x^{2} + 9)}^{2}}$

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

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

Add $1$ and $2$ .

$f' (x) = \frac{2 x^{3} + 2 \cdot (9 x) - 2 x^{3}}{{(x^{2} + 9)}^{2}}$

Multiply $2$ by $9$ .

$f' (x) = \frac{2 x^{3} + 18 x - 2 x^{3}}{{(x^{2} + 9)}^{2}}$

Combine the opposite terms in $2 x^{3} + 18 x - 2 x^{3}$ .

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

$f' (x) = \frac{18 x + 0}{{(x^{2} + 9)}^{2}}$

Add $18 x$ and $0$ .

$f' (x) = \frac{18 x}{{(x^{2} + 9)}^{2}}$

The first derivative of $f (x)$ with respect to $x$ is $\frac{18 x}{{(x^{2} + 9)}^{2}}$ .

$\frac{18 x}{{(x^{2} + 9)}^{2}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{18 x}{{(x^{2} + 9)}^{2}} = 0$ .

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Set the first derivative equal to $0$ .

$\frac{18 x}{{(x^{2} + 9)}^{2}} = 0$

Set the numerator equal to zero.

$18 x = 0$

Divide each term in $18 x = 0$ by $18$ and simplify.

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Divide each term in $18 x = 0$ by $18$ .

$\frac{18 x}{18} = \frac{0}{18}$

Simplify the left side.

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Cancel the common factor of $18$ .

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Cancel the common factor.

$\frac{18 x}{18} = \frac{0}{18}$

Divide $x$ by $1$ .

$x = \frac{0}{18}$

Simplify the right side.

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Divide $0$ by $18$ .

$x = 0$

Step 7

Find the values where the derivative is undefined.

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The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 8

Critical points to evaluate.

$x = 0$

Step 9

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{54 ({(0)}^{2} - 3)}{{({(0)}^{2} + 9)}^{3}}$

Step 10

Evaluate the second derivative.

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Simplify the numerator.

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

$- \frac{54 (0 - 3)}{{(0^{2} + 9)}^{3}}$

Subtract $3$ from $0$ .

$- \frac{54 \cdot - 3}{{(0^{2} + 9)}^{3}}$

Simplify the denominator.

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

$- \frac{54 \cdot - 3}{{(0 + 9)}^{3}}$

Add $0$ and $9$ .

$- \frac{54 \cdot - 3}{9^{3}}$

Raise $9$ to the power of $3$ .

$- \frac{54 \cdot - 3}{729}$

Reduce the expression by cancelling the common factors.

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Multiply $54$ by $- 3$ .

$- \frac{- 162}{729}$

Cancel the common factor of $- 162$ and $729$ .

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Factor $81$ out of $- 162$ .

$- \frac{81 (- 2)}{729}$

Cancel the common factors.

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Factor $81$ out of $729$ .

$- \frac{81 \cdot - 2}{81 \cdot 9}$

Cancel the common factor.

$- \frac{81 \cdot - 2}{81 \cdot 9}$

Rewrite the expression.

$- \frac{- 2}{9}$

Move the negative in front of the fraction.

$- - \frac{2}{9}$

Multiply $- - \frac{2}{9}$ .

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Multiply $- 1$ by $- 1$ .

$1 (\frac{2}{9})$

Multiply $\frac{2}{9}$ by $1$ .

$\frac{2}{9}$

Step 11

$x = 0$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 0$ is a local minimum

Step 12

Find the y-value when $x = 0$ .

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

$f (0) = \frac{{(0)}^{2}}{{(0)}^{2} + 9}$

Simplify the result.

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

$f (0) = \frac{0}{0^{2} + 9}$

Simplify the denominator.

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

$f (0) = \frac{0}{0 + 9}$

Add $0$ and $9$ .

$f (0) = \frac{0}{9}$

Divide $0$ by $9$ .

$f (0) = 0$

The final answer is $0$ .

$y = 0$

Step 13

These are the local extrema for $f (x) = \frac{x^{2}}{x^{2} + 9}$ .

$(0, 0)$ is a local minima

Step 14