Calculus Examples

$y = \frac{x^{2}}{x^{2} + 108}$

Step 1

Write $y = \frac{x^{2}}{x^{2} + 108}$ as a function.

$f (x) = \frac{x^{2}}{x^{2} + 108}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

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 $g (x) = x^{2} + 108$ .

$\frac{(x^{2} + 108) \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [x^{2} + 108]}{{(x^{2} + 108)}^{2}}$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

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

$\frac{(x^{2} + 108) (2 x) - x^{2} \frac{d}{d x} [x^{2} + 108]}{{(x^{2} + 108)}^{2}}$

Step 2.1.2.2

Move $2$ to the left of $x^{2} + 108$ .

$\frac{2 \cdot (x^{2} + 108) x - x^{2} \frac{d}{d x} [x^{2} + 108]}{{(x^{2} + 108)}^{2}}$

Step 2.1.2.3

By the Sum Rule, the derivative of $x^{2} + 108$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [108]$ .

$\frac{2 (x^{2} + 108) x - x^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [108])}{{(x^{2} + 108)}^{2}}$

Step 2.1.2.4

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

$\frac{2 (x^{2} + 108) x - x^{2} (2 x + \frac{d}{d x} [108])}{{(x^{2} + 108)}^{2}}$

Step 2.1.2.5

Since $108$ is constant with respect to $x$ , the derivative of $108$ with respect to $x$ is $0$ .

$\frac{2 (x^{2} + 108) x - x^{2} (2 x + 0)}{{(x^{2} + 108)}^{2}}$

Step 2.1.2.6

Simplify the expression.

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Step 2.1.2.6.1

Add $2 x$ and $0$ .

$\frac{2 (x^{2} + 108) x - x^{2} (2 x)}{{(x^{2} + 108)}^{2}}$

Step 2.1.2.6.2

Multiply $2$ by $- 1$ .

$\frac{2 (x^{2} + 108) x - 2 x^{2} x}{{(x^{2} + 108)}^{2}}$

Step 2.1.3

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

$\frac{2 (x^{2} + 108) x - 2 (x^{1} x^{2})}{{(x^{2} + 108)}^{2}}$

Step 2.1.4

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

$\frac{2 (x^{2} + 108) x - 2 x^{1 + 2}}{{(x^{2} + 108)}^{2}}$

Step 2.1.5

Add $1$ and $2$ .

$\frac{2 (x^{2} + 108) x - 2 x^{3}}{{(x^{2} + 108)}^{2}}$

Step 2.1.6

Simplify.

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Step 2.1.6.1

Apply the distributive property.

$\frac{(2 x^{2} + 2 \cdot 108) x - 2 x^{3}}{{(x^{2} + 108)}^{2}}$

Step 2.1.6.2

Apply the distributive property.

$\frac{2 x^{2} x + 2 \cdot 108 x - 2 x^{3}}{{(x^{2} + 108)}^{2}}$

Step 2.1.6.3

Simplify the numerator.

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Step 2.1.6.3.1

Simplify each term.

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Step 2.1.6.3.1.1

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

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Step 2.1.6.3.1.1.1

Move $x$ .

$\frac{2 (x \cdot x^{2}) + 2 \cdot 108 x - 2 x^{3}}{{(x^{2} + 108)}^{2}}$

Step 2.1.6.3.1.1.2

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

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Step 2.1.6.3.1.1.2.1

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

$\frac{2 (x^{1} x^{2}) + 2 \cdot 108 x - 2 x^{3}}{{(x^{2} + 108)}^{2}}$

Step 2.1.6.3.1.1.2.2

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

$\frac{2 x^{1 + 2} + 2 \cdot 108 x - 2 x^{3}}{{(x^{2} + 108)}^{2}}$

Step 2.1.6.3.1.1.3

Add $1$ and $2$ .

$\frac{2 x^{3} + 2 \cdot 108 x - 2 x^{3}}{{(x^{2} + 108)}^{2}}$

Step 2.1.6.3.1.2

Multiply $2$ by $108$ .

$\frac{2 x^{3} + 216 x - 2 x^{3}}{{(x^{2} + 108)}^{2}}$

Step 2.1.6.3.2

Combine the opposite terms in $2 x^{3} + 216 x - 2 x^{3}$ .

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Step 2.1.6.3.2.1

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

$\frac{216 x + 0}{{(x^{2} + 108)}^{2}}$

Step 2.1.6.3.2.2

Add $216 x$ and $0$ .

$f' (x) = \frac{216 x}{{(x^{2} + 108)}^{2}}$

Step 2.2

Find the second derivative.

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Step 2.2.1

Since $216$ is constant with respect to $x$ , the derivative of $\frac{216 x}{{(x^{2} + 108)}^{2}}$ with respect to $x$ is $216 \frac{d}{d x} [\frac{x}{{(x^{2} + 108)}^{2}}]$ .

$216 \frac{d}{d x} [\frac{x}{{(x^{2} + 108)}^{2}}]$

Step 2.2.2

$216 \frac{{(x^{2} + 108)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} + 108)}^{2}]}{{({(x^{2} + 108)}^{2})}^{2}}$

Step 2.2.3

Differentiate using the Power Rule.

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Step 2.2.3.1

Multiply the exponents in ${({(x^{2} + 108)}^{2})}^{2}$ .

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Step 2.2.3.1.1

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

$216 \frac{{(x^{2} + 108)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} + 108)}^{2}]}{{(x^{2} + 108)}^{2 \cdot 2}}$

Step 2.2.3.1.2

Multiply $2$ by $2$ .

$216 \frac{{(x^{2} + 108)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} + 108)}^{2}]}{{(x^{2} + 108)}^{4}}$

Step 2.2.3.2

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

$216 \frac{{(x^{2} + 108)}^{2} \cdot 1 - x \frac{d}{d x} [{(x^{2} + 108)}^{2}]}{{(x^{2} + 108)}^{4}}$

Step 2.2.3.3

Multiply ${(x^{2} + 108)}^{2}$ by $1$ .

$216 \frac{{(x^{2} + 108)}^{2} - x \frac{d}{d x} [{(x^{2} + 108)}^{2}]}{{(x^{2} + 108)}^{4}}$

Step 2.2.4

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 $g (x) = x^{2} + 108$ .

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Step 2.2.4.1

To apply the Chain Rule, set $u$ as $x^{2} + 108$ .

$216 \frac{{(x^{2} + 108)}^{2} - x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} + 108])}{{(x^{2} + 108)}^{4}}$

Step 2.2.4.2

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

$216 \frac{{(x^{2} + 108)}^{2} - x (2 u \frac{d}{d x} [x^{2} + 108])}{{(x^{2} + 108)}^{4}}$

Step 2.2.4.3

Replace all occurrences of $u$ with $x^{2} + 108$ .

$216 \frac{{(x^{2} + 108)}^{2} - x (2 (x^{2} + 108) \frac{d}{d x} [x^{2} + 108])}{{(x^{2} + 108)}^{4}}$

Step 2.2.5

Simplify with factoring out.

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Step 2.2.5.1

Multiply $2$ by $- 1$ .

$216 \frac{{(x^{2} + 108)}^{2} - 2 x ((x^{2} + 108) \frac{d}{d x} [x^{2} + 108])}{{(x^{2} + 108)}^{4}}$

Step 2.2.5.2

Factor $x^{2} + 108$ out of ${(x^{2} + 108)}^{2} - 2 x ((x^{2} + 108) \frac{d}{d x} [x^{2} + 108])$ .

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Step 2.2.5.2.1

Factor $x^{2} + 108$ out of ${(x^{2} + 108)}^{2}$ .

$216 \frac{(x^{2} + 108) (x^{2} + 108) - 2 x ((x^{2} + 108) \frac{d}{d x} [x^{2} + 108])}{{(x^{2} + 108)}^{4}}$

Step 2.2.5.2.2

Factor $x^{2} + 108$ out of $- 2 x ((x^{2} + 108) \frac{d}{d x} [x^{2} + 108])$ .

$216 \frac{(x^{2} + 108) (x^{2} + 108) + (x^{2} + 108) (- 2 x (\frac{d}{d x} [x^{2} + 108]))}{{(x^{2} + 108)}^{4}}$

Step 2.2.5.2.3

Factor $x^{2} + 108$ out of $(x^{2} + 108) (x^{2} + 108) + (x^{2} + 108) (- 2 x (\frac{d}{d x} [x^{2} + 108]))$ .

$216 \frac{(x^{2} + 108) (x^{2} + 108 - 2 x (\frac{d}{d x} [x^{2} + 108]))}{{(x^{2} + 108)}^{4}}$

Step 2.2.6

Cancel the common factors.

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Step 2.2.6.1

Factor $x^{2} + 108$ out of ${(x^{2} + 108)}^{4}$ .

$216 \frac{(x^{2} + 108) (x^{2} + 108 - 2 x (\frac{d}{d x} [x^{2} + 108]))}{(x^{2} + 108) {(x^{2} + 108)}^{3}}$

Step 2.2.6.2

Cancel the common factor.

$216 \frac{(x^{2} + 108) (x^{2} + 108 - 2 x (\frac{d}{d x} [x^{2} + 108]))}{(x^{2} + 108) {(x^{2} + 108)}^{3}}$

Step 2.2.6.3

Rewrite the expression.

$216 \frac{x^{2} + 108 - 2 x (\frac{d}{d x} [x^{2} + 108])}{{(x^{2} + 108)}^{3}}$

Step 2.2.7

By the Sum Rule, the derivative of $x^{2} + 108$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [108]$ .

$216 \frac{x^{2} + 108 - 2 x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [108])}{{(x^{2} + 108)}^{3}}$

Step 2.2.8

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

$216 \frac{x^{2} + 108 - 2 x (2 x + \frac{d}{d x} [108])}{{(x^{2} + 108)}^{3}}$

Step 2.2.9

Since $108$ is constant with respect to $x$ , the derivative of $108$ with respect to $x$ is $0$ .

$216 \frac{x^{2} + 108 - 2 x (2 x + 0)}{{(x^{2} + 108)}^{3}}$

Step 2.2.10

Simplify the expression.

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Step 2.2.10.1

Add $2 x$ and $0$ .

$216 \frac{x^{2} + 108 - 2 x (2 x)}{{(x^{2} + 108)}^{3}}$

Step 2.2.10.2

Multiply $2$ by $- 2$ .

$216 \frac{x^{2} + 108 - 4 x \cdot x}{{(x^{2} + 108)}^{3}}$

Step 2.2.11

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

$216 \frac{x^{2} + 108 - 4 (x^{1} x)}{{(x^{2} + 108)}^{3}}$

Step 2.2.12

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

$216 \frac{x^{2} + 108 - 4 (x^{1} x^{1})}{{(x^{2} + 108)}^{3}}$

Step 2.2.13

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

$216 \frac{x^{2} + 108 - 4 x^{1 + 1}}{{(x^{2} + 108)}^{3}}$

Step 2.2.14

Add $1$ and $1$ .

$216 \frac{x^{2} + 108 - 4 x^{2}}{{(x^{2} + 108)}^{3}}$

Step 2.2.15

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

$216 \frac{- 3 x^{2} + 108}{{(x^{2} + 108)}^{3}}$

Step 2.2.16

Combine $216$ and $\frac{- 3 x^{2} + 108}{{(x^{2} + 108)}^{3}}$ .

$\frac{216 (- 3 x^{2} + 108)}{{(x^{2} + 108)}^{3}}$

Step 2.2.17

Simplify.

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Step 2.2.17.1

Apply the distributive property.

$\frac{216 (- 3 x^{2}) + 216 \cdot 108}{{(x^{2} + 108)}^{3}}$

Step 2.2.17.2

Simplify each term.

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Step 2.2.17.2.1

Multiply $- 3$ by $216$ .

$\frac{- 648 x^{2} + 216 \cdot 108}{{(x^{2} + 108)}^{3}}$

Step 2.2.17.2.2

Multiply $216$ by $108$ .

$\frac{- 648 x^{2} + 23328}{{(x^{2} + 108)}^{3}}$

Step 2.2.17.3

Simplify the numerator.

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Step 2.2.17.3.1

Factor $648$ out of $- 648 x^{2} + 23328$ .

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Step 2.2.17.3.1.1

Factor $648$ out of $- 648 x^{2}$ .

$\frac{648 (- x^{2}) + 23328}{{(x^{2} + 108)}^{3}}$

Step 2.2.17.3.1.2

Factor $648$ out of $23328$ .

$\frac{648 (- x^{2}) + 648 (36)}{{(x^{2} + 108)}^{3}}$

Step 2.2.17.3.1.3

Factor $648$ out of $648 (- x^{2}) + 648 (36)$ .

$\frac{648 (- x^{2} + 36)}{{(x^{2} + 108)}^{3}}$

Step 2.2.17.3.2

Rewrite $36$ as $6^{2}$ .

$\frac{648 (- x^{2} + 6^{2})}{{(x^{2} + 108)}^{3}}$

Step 2.2.17.3.3

Reorder $- x^{2}$ and $6^{2}$ .

$\frac{648 (6^{2} - x^{2})}{{(x^{2} + 108)}^{3}}$

Step 2.2.17.3.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6$ and $b = x$ .

$f'' (x) = \frac{648 (6 + x) (6 - x)}{{(x^{2} + 108)}^{3}}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{648 (6 + x) (6 - x)}{{(x^{2} + 108)}^{3}}$ .

$\frac{648 (6 + x) (6 - x)}{{(x^{2} + 108)}^{3}}$

Step 3

Set the second derivative equal to $0$ then solve the equation $\frac{648 (6 + x) (6 - x)}{{(x^{2} + 108)}^{3}} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$\frac{648 (6 + x) (6 - x)}{{(x^{2} + 108)}^{3}} = 0$

Step 3.2

Set the numerator equal to zero.

$648 (6 + x) (6 - x) = 0$

Step 3.3

Solve the equation for $x$ .

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Step 3.3.1

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

$6 + x = 0$

$6 - x = 0$

Step 3.3.2

Set $6 + x$ equal to $0$ and solve for $x$ .

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Step 3.3.2.1

Set $6 + x$ equal to $0$ .

$6 + x = 0$

Step 3.3.2.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 3.3.3

Set $6 - x$ equal to $0$ and solve for $x$ .

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Step 3.3.3.1

Set $6 - x$ equal to $0$ .

$6 - x = 0$

Step 3.3.3.2

Solve $6 - x = 0$ for $x$ .

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Step 3.3.3.2.1

Subtract $6$ from both sides of the equation.

$- x = - 6$

Step 3.3.3.2.2

Divide each term in $- x = - 6$ by $- 1$ and simplify.

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Step 3.3.3.2.2.1

Divide each term in $- x = - 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 6}{- 1}$

Step 3.3.3.2.2.2

Simplify the left side.

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Step 3.3.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 6}{- 1}$

Step 3.3.3.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 6}{- 1}$

Step 3.3.3.2.2.3

Simplify the right side.

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Step 3.3.3.2.2.3.1

Divide $- 6$ by $- 1$ .

$x = 6$

Step 3.3.4

The final solution is all the values that make $648 (6 + x) (6 - x) = 0$ true.

$x = - 6, 6$

Step 4

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

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Step 4.1

Substitute $- 6$ in $f (x) = \frac{x^{2}}{x^{2} + 108}$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $- 6$ in the expression.

$f (- 6) = \frac{{(- 6)}^{2}}{{(- 6)}^{2} + 108}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Raise $- 6$ to the power of $2$ .

$f (- 6) = \frac{36}{{(- 6)}^{2} + 108}$

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

Raise $- 6$ to the power of $2$ .

$f (- 6) = \frac{36}{36 + 108}$

Step 4.1.2.2.2

Add $36$ and $108$ .

$f (- 6) = \frac{36}{144}$

Step 4.1.2.3

Cancel the common factor of $36$ and $144$ .

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Step 4.1.2.3.1

Factor $36$ out of $36$ .

$f (- 6) = \frac{36 (1)}{144}$

Step 4.1.2.3.2

Cancel the common factors.

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Step 4.1.2.3.2.1

Factor $36$ out of $144$ .

$f (- 6) = \frac{36 \cdot 1}{36 \cdot 4}$

Step 4.1.2.3.2.2

Cancel the common factor.

$f (- 6) = \frac{36 \cdot 1}{36 \cdot 4}$

Step 4.1.2.3.2.3

Rewrite the expression.

$f (- 6) = \frac{1}{4}$

Step 4.1.2.4

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

$\frac{1}{4}$

Step 4.2

The point found by substituting $- 6$ in $f (x) = \frac{x^{2}}{x^{2} + 108}$ is $(- 6, \frac{1}{4})$ . This point can be an inflection point.

$(- 6, \frac{1}{4})$

Step 4.3

Substitute $6$ in $f (x) = \frac{x^{2}}{x^{2} + 108}$ to find the value of $y$ .

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Step 4.3.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = \frac{{(6)}^{2}}{{(6)}^{2} + 108}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Raise $6$ to the power of $2$ .

$f (6) = \frac{36}{6^{2} + 108}$

Step 4.3.2.2

Simplify the denominator.

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Step 4.3.2.2.1

Raise $6$ to the power of $2$ .

$f (6) = \frac{36}{36 + 108}$

Step 4.3.2.2.2

Add $36$ and $108$ .

$f (6) = \frac{36}{144}$

Step 4.3.2.3

Cancel the common factor of $36$ and $144$ .

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Step 4.3.2.3.1

Factor $36$ out of $36$ .

$f (6) = \frac{36 (1)}{144}$

Step 4.3.2.3.2

Cancel the common factors.

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Step 4.3.2.3.2.1

Factor $36$ out of $144$ .

$f (6) = \frac{36 \cdot 1}{36 \cdot 4}$

Step 4.3.2.3.2.2

Cancel the common factor.

$f (6) = \frac{36 \cdot 1}{36 \cdot 4}$

Step 4.3.2.3.2.3

Rewrite the expression.

$f (6) = \frac{1}{4}$

Step 4.3.2.4

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

$\frac{1}{4}$

Step 4.4

The point found by substituting $6$ in $f (x) = \frac{x^{2}}{x^{2} + 108}$ is $(6, \frac{1}{4})$ . This point can be an inflection point.

$(6, \frac{1}{4})$

Step 4.5

Determine the points that could be inflection points.

$(- 6, \frac{1}{4}), (6, \frac{1}{4})$

Step 5

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

$(- \infty, - 6) \cup (- 6, 6) \cup (6, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 6)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 6.1$ in the expression.

$f'' (- 6.1) = \frac{648 (6 - 6.1) (6 - (- 6.1))}{{({(- 6.1)}^{2} + 108)}^{3}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Remove parentheses.

$f'' (- 6.1) = \frac{648 (6 - 6.1) (6 - (- 6.1))}{{({(- 6.1)}^{2} + 108)}^{3}}$

Step 6.2.2

Simplify the numerator.

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Step 6.2.2.1

Multiply $- 1$ by $- 6.1$ .

$f'' (- 6.1) = \frac{648 (6 - 6.1) (6 + 6.1)}{{({(- 6.1)}^{2} + 108)}^{3}}$

Step 6.2.2.2

Subtract $6.1$ from $6$ .

$f'' (- 6.1) = \frac{648 \cdot (- 0.1 (6 + 6.1))}{{({(- 6.1)}^{2} + 108)}^{3}}$

Step 6.2.2.3

Combine exponents.

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Step 6.2.2.3.1

Multiply $648$ by $- 0.1$ .

$f'' (- 6.1) = \frac{- 64.8 (6 + 6.1)}{{({(- 6.1)}^{2} + 108)}^{3}}$

Step 6.2.2.3.2

Multiply $- 64.8$ by $6 + 6.1$ .

$f'' (- 6.1) = \frac{- 784.08}{{({(- 6.1)}^{2} + 108)}^{3}}$

Step 6.2.3

Simplify the denominator.

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Step 6.2.3.1

Raise $- 6.1$ to the power of $2$ .

$f'' (- 6.1) = \frac{- 784.08}{{(37.21 + 108)}^{3}}$

Step 6.2.3.2

Add $37.21$ and $108$ .

$f'' (- 6.1) = \frac{- 784.08}{{145.21}^{3}}$

Step 6.2.3.3

Raise $145.21$ to the power of $3$ .

$f'' (- 6.1) = \frac{- 784.08}{3061889.942761}$

Step 6.2.4

Divide $- 784.08$ by $3061889.942761$ .

$f'' (- 6.1) = - 0.00025607$

Step 6.2.5

The final answer is $- 0.00025607$ .

$- 0.00025607$

Step 6.3

At $- 6.1$ , the second derivative is $- 0.00025607$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 6)$

Decreasing on $(- \infty, - 6)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(- 6, 6)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

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

$f'' (0) = \frac{648 (6 + 0) (6 - (0))}{{({(0)}^{2} + 108)}^{3}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Remove parentheses.

$f'' (0) = \frac{648 (6 + 0) (6 - (0))}{{({(0)}^{2} + 108)}^{3}}$

Step 7.2.2

Simplify the numerator.

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Step 7.2.2.1

Combine exponents.

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Step 7.2.2.1.1

Multiply $- 1$ by $0$ .

$f'' (0) = \frac{648 (6 + 0) (6 + 0)}{{(0^{2} + 108)}^{3}}$

Step 7.2.2.1.2

Raise $6 + 0$ to the power of $1$ .

$f'' (0) = \frac{648 ((6 + 0) (6 + 0))}{{(0^{2} + 108)}^{3}}$

Step 7.2.2.1.3

Raise $6 + 0$ to the power of $1$ .

$f'' (0) = \frac{648 ((6 + 0) (6 + 0))}{{(0^{2} + 108)}^{3}}$

Step 7.2.2.1.4

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

$f'' (0) = \frac{648 {(6 + 0)}^{1 + 1}}{{(0^{2} + 108)}^{3}}$

Step 7.2.2.1.5

Add $1$ and $1$ .

$f'' (0) = \frac{648 {(6 + 0)}^{2}}{{(0^{2} + 108)}^{3}}$

Step 7.2.2.2

Add $6$ and $0$ .

$f'' (0) = \frac{648 \cdot 6^{2}}{{(0^{2} + 108)}^{3}}$

Step 7.2.2.3

Raise $6$ to the power of $2$ .

$f'' (0) = \frac{648 \cdot 36}{{(0^{2} + 108)}^{3}}$

Step 7.2.3

Simplify the denominator.

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Step 7.2.3.1

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

$f'' (0) = \frac{648 \cdot 36}{{(0 + 108)}^{3}}$

Step 7.2.3.2

Add $0$ and $108$ .

$f'' (0) = \frac{648 \cdot 36}{108^{3}}$

Step 7.2.3.3

Raise $108$ to the power of $3$ .

$f'' (0) = \frac{648 \cdot 36}{1259712}$

Step 7.2.4

Reduce the expression by cancelling the common factors.

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Step 7.2.4.1

Multiply $648$ by $36$ .

$f'' (0) = \frac{23328}{1259712}$

Step 7.2.4.2

Cancel the common factor of $23328$ and $1259712$ .

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Step 7.2.4.2.1

Factor $23328$ out of $23328$ .

$f'' (0) = \frac{23328 (1)}{1259712}$

Step 7.2.4.2.2

Cancel the common factors.

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Step 7.2.4.2.2.1

Factor $23328$ out of $1259712$ .

$f'' (0) = \frac{23328 \cdot 1}{23328 \cdot 54}$

Step 7.2.4.2.2.2

Cancel the common factor.

$f'' (0) = \frac{23328 \cdot 1}{23328 \cdot 54}$

Step 7.2.4.2.2.3

Rewrite the expression.

$f'' (0) = \frac{1}{54}$

Step 7.2.5

The final answer is $\frac{1}{54}$ .

$\frac{1}{54}$

Step 7.3

At $0$ , the second derivative is $0.0\overline{185}$ . Since this is positive, the second derivative is increasing on the interval $(- 6, 6)$ .

Increasing on $(- 6, 6)$ since $f'' (x) > 0$

Step 8

Substitute a value from the interval $(6, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $6.1$ in the expression.

$f'' (6.1) = \frac{648 (6 + 6.1) (6 - (6.1))}{{({(6.1)}^{2} + 108)}^{3}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Remove parentheses.

$f'' (6.1) = \frac{648 (6 + 6.1) (6 - (6.1))}{{({(6.1)}^{2} + 108)}^{3}}$

Step 8.2.2

Simplify the numerator.

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Step 8.2.2.1

Multiply $- 1$ by $6.1$ .

$f'' (6.1) = \frac{648 (6 + 6.1) (6 - 6.1)}{{({6.1}^{2} + 108)}^{3}}$

Step 8.2.2.2

Add $6$ and $6.1$ .

$f'' (6.1) = \frac{648 \cdot (12.1 (6 - 6.1))}{{({6.1}^{2} + 108)}^{3}}$

Step 8.2.2.3

Combine exponents.

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Step 8.2.2.3.1

Multiply $648$ by $12.1$ .

$f'' (6.1) = \frac{7840.8 (6 - 6.1)}{{({6.1}^{2} + 108)}^{3}}$

Step 8.2.2.3.2

Multiply $7840.8$ by $6 - 6.1$ .

$f'' (6.1) = \frac{- 784.07999999}{{({6.1}^{2} + 108)}^{3}}$

Step 8.2.3

Simplify the denominator.

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Step 8.2.3.1

Raise $6.1$ to the power of $2$ .

$f'' (6.1) = \frac{- 784.07999999}{{(37.21 + 108)}^{3}}$

Step 8.2.3.2

Add $37.21$ and $108$ .

$f'' (6.1) = \frac{- 784.07999999}{{145.21}^{3}}$

Step 8.2.3.3

Raise $145.21$ to the power of $3$ .

$f'' (6.1) = \frac{- 784.07999999}{3061889.942761}$

Step 8.2.4

Divide $- 784.07999999$ by $3061889.942761$ .

$f'' (6.1) = - 0.00025607$

Step 8.2.5

The final answer is $- 0.00025607$ .

$- 0.00025607$

Step 8.3

At $6.1$ , the second derivative is $- 0.00025607$ . Since this is negative, the second derivative is decreasing on the interval $(6, \infty)$

Decreasing on $(6, \infty)$ since $f'' (x) < 0$

Step 9

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 $(- 6, \frac{1}{4}), (6, \frac{1}{4})$ .

$(- 6, \frac{1}{4}), (6, \frac{1}{4})$

Step 10