Calculus Examples

$f (x) = \frac{1}{x^{2} + 7}$

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

Rewrite $\frac{1}{x^{2} + 7}$ as ${(x^{2} + 7)}^{- 1}$ .

$\frac{d}{d x} [{(x^{2} + 7)}^{- 1}]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2} + 7$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 7$ .

$\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} + 7]$

Step 1.1.2.2

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

$- u^{- 2} \frac{d}{d x} [x^{2} + 7]$

Step 1.1.2.3

Replace all occurrences of $u$ with $x^{2} + 7$ .

$- {(x^{2} + 7)}^{- 2} \frac{d}{d x} [x^{2} + 7]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

By the Sum Rule, the derivative of $x^{2} + 7$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [7]$ .

$- {(x^{2} + 7)}^{- 2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [7])$

Step 1.1.3.2

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

$- {(x^{2} + 7)}^{- 2} (2 x + \frac{d}{d x} [7])$

Step 1.1.3.3

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$- {(x^{2} + 7)}^{- 2} (2 x + 0)$

Step 1.1.3.4

Simplify the expression.

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Step 1.1.3.4.1

Add $2 x$ and $0$ .

$- {(x^{2} + 7)}^{- 2} (2 x)$

Step 1.1.3.4.2

Multiply $2$ by $- 1$ .

$- 2 {(x^{2} + 7)}^{- 2} x$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 2 \frac{1}{{(x^{2} + 7)}^{2}} x$

Step 1.1.4.2

Combine terms.

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Step 1.1.4.2.1

Combine $- 2$ and $\frac{1}{{(x^{2} + 7)}^{2}}$ .

$\frac{- 2}{{(x^{2} + 7)}^{2}} x$

Step 1.1.4.2.2

Move the negative in front of the fraction.

$- \frac{2}{{(x^{2} + 7)}^{2}} x$

Step 1.1.4.2.3

Combine $x$ and $\frac{2}{{(x^{2} + 7)}^{2}}$ .

$- \frac{x \cdot 2}{{(x^{2} + 7)}^{2}}$

Step 1.1.4.2.4

Move $2$ to the left of $x$ .

$f' (x) = - \frac{2 x}{{(x^{2} + 7)}^{2}}$

Step 1.2

Find the second derivative.

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Step 1.2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2 x}{{(x^{2} + 7)}^{2}}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{x}{{(x^{2} + 7)}^{2}}]$ .

$- 2 \frac{d}{d x} [\frac{x}{{(x^{2} + 7)}^{2}}]$

Step 1.2.2

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 $f (x) = x$ and $g (x) = {(x^{2} + 7)}^{2}$ .

$- 2 \frac{{(x^{2} + 7)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} + 7)}^{2}]}{{({(x^{2} + 7)}^{2})}^{2}}$

Step 1.2.3

Differentiate using the Power Rule.

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Step 1.2.3.1

Multiply the exponents in ${({(x^{2} + 7)}^{2})}^{2}$ .

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Step 1.2.3.1.1

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

$- 2 \frac{{(x^{2} + 7)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} + 7)}^{2}]}{{(x^{2} + 7)}^{2 \cdot 2}}$

Step 1.2.3.1.2

Multiply $2$ by $2$ .

$- 2 \frac{{(x^{2} + 7)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} + 7)}^{2}]}{{(x^{2} + 7)}^{4}}$

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 $n = 1$ .

$- 2 \frac{{(x^{2} + 7)}^{2} \cdot 1 - x \frac{d}{d x} [{(x^{2} + 7)}^{2}]}{{(x^{2} + 7)}^{4}}$

Step 1.2.3.3

Multiply ${(x^{2} + 7)}^{2}$ by $1$ .

$- 2 \frac{{(x^{2} + 7)}^{2} - x \frac{d}{d x} [{(x^{2} + 7)}^{2}]}{{(x^{2} + 7)}^{4}}$

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 $g (x) = x^{2} + 7$ .

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Step 1.2.4.1

To apply the Chain Rule, set $u$ as $x^{2} + 7$ .

$- 2 \frac{{(x^{2} + 7)}^{2} - x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} + 7])}{{(x^{2} + 7)}^{4}}$

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 $n = 2$ .

$- 2 \frac{{(x^{2} + 7)}^{2} - x (2 u \frac{d}{d x} [x^{2} + 7])}{{(x^{2} + 7)}^{4}}$

Step 1.2.4.3

Replace all occurrences of $u$ with $x^{2} + 7$ .

$- 2 \frac{{(x^{2} + 7)}^{2} - x (2 (x^{2} + 7) \frac{d}{d x} [x^{2} + 7])}{{(x^{2} + 7)}^{4}}$

Step 1.2.5

Simplify with factoring out.

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Step 1.2.5.1

Multiply $2$ by $- 1$ .

$- 2 \frac{{(x^{2} + 7)}^{2} - 2 x ((x^{2} + 7) \frac{d}{d x} [x^{2} + 7])}{{(x^{2} + 7)}^{4}}$

Step 1.2.5.2

Factor $x^{2} + 7$ out of ${(x^{2} + 7)}^{2} - 2 x ((x^{2} + 7) \frac{d}{d x} [x^{2} + 7])$ .

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Step 1.2.5.2.1

Factor $x^{2} + 7$ out of ${(x^{2} + 7)}^{2}$ .

$- 2 \frac{(x^{2} + 7) (x^{2} + 7) - 2 x ((x^{2} + 7) \frac{d}{d x} [x^{2} + 7])}{{(x^{2} + 7)}^{4}}$

Step 1.2.5.2.2

Factor $x^{2} + 7$ out of $- 2 x ((x^{2} + 7) \frac{d}{d x} [x^{2} + 7])$ .

$- 2 \frac{(x^{2} + 7) (x^{2} + 7) + (x^{2} + 7) (- 2 x (\frac{d}{d x} [x^{2} + 7]))}{{(x^{2} + 7)}^{4}}$

Step 1.2.5.2.3

Factor $x^{2} + 7$ out of $(x^{2} + 7) (x^{2} + 7) + (x^{2} + 7) (- 2 x (\frac{d}{d x} [x^{2} + 7]))$ .

$- 2 \frac{(x^{2} + 7) (x^{2} + 7 - 2 x (\frac{d}{d x} [x^{2} + 7]))}{{(x^{2} + 7)}^{4}}$

Step 1.2.6

Cancel the common factors.

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Step 1.2.6.1

Factor $x^{2} + 7$ out of ${(x^{2} + 7)}^{4}$ .

$- 2 \frac{(x^{2} + 7) (x^{2} + 7 - 2 x (\frac{d}{d x} [x^{2} + 7]))}{(x^{2} + 7) {(x^{2} + 7)}^{3}}$

Step 1.2.6.2

Cancel the common factor.

$- 2 \frac{(x^{2} + 7) (x^{2} + 7 - 2 x (\frac{d}{d x} [x^{2} + 7]))}{(x^{2} + 7) {(x^{2} + 7)}^{3}}$

Step 1.2.6.3

Rewrite the expression.

$- 2 \frac{x^{2} + 7 - 2 x (\frac{d}{d x} [x^{2} + 7])}{{(x^{2} + 7)}^{3}}$

Step 1.2.7

By the Sum Rule, the derivative of $x^{2} + 7$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [7]$ .

$- 2 \frac{x^{2} + 7 - 2 x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [7])}{{(x^{2} + 7)}^{3}}$

Step 1.2.8

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

$- 2 \frac{x^{2} + 7 - 2 x (2 x + \frac{d}{d x} [7])}{{(x^{2} + 7)}^{3}}$

Step 1.2.9

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$- 2 \frac{x^{2} + 7 - 2 x (2 x + 0)}{{(x^{2} + 7)}^{3}}$

Step 1.2.10

Simplify the expression.

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Step 1.2.10.1

Add $2 x$ and $0$ .

$- 2 \frac{x^{2} + 7 - 2 x (2 x)}{{(x^{2} + 7)}^{3}}$

Step 1.2.10.2

Multiply $2$ by $- 2$ .

$- 2 \frac{x^{2} + 7 - 4 x \cdot x}{{(x^{2} + 7)}^{3}}$

Step 1.2.11

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

$- 2 \frac{x^{2} + 7 - 4 (x^{1} x)}{{(x^{2} + 7)}^{3}}$

Step 1.2.12

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

$- 2 \frac{x^{2} + 7 - 4 (x^{1} x^{1})}{{(x^{2} + 7)}^{3}}$

Step 1.2.13

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

$- 2 \frac{x^{2} + 7 - 4 x^{1 + 1}}{{(x^{2} + 7)}^{3}}$

Step 1.2.14

Add $1$ and $1$ .

$- 2 \frac{x^{2} + 7 - 4 x^{2}}{{(x^{2} + 7)}^{3}}$

Step 1.2.15

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

$- 2 \frac{- 3 x^{2} + 7}{{(x^{2} + 7)}^{3}}$

Step 1.2.16

Combine $- 2$ and $\frac{- 3 x^{2} + 7}{{(x^{2} + 7)}^{3}}$ .

$\frac{- 2 (- 3 x^{2} + 7)}{{(x^{2} + 7)}^{3}}$

Step 1.2.17

Move the negative in front of the fraction.

$- \frac{2 (- 3 x^{2} + 7)}{{(x^{2} + 7)}^{3}}$

Step 1.2.18

Simplify.

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Step 1.2.18.1

Apply the distributive property.

$- \frac{2 (- 3 x^{2}) + 2 \cdot 7}{{(x^{2} + 7)}^{3}}$

Step 1.2.18.2

Simplify each term.

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Step 1.2.18.2.1

Multiply $- 3$ by $2$ .

$- \frac{- 6 x^{2} + 2 \cdot 7}{{(x^{2} + 7)}^{3}}$

Step 1.2.18.2.2

Multiply $2$ by $7$ .

$- \frac{- 6 x^{2} + 14}{{(x^{2} + 7)}^{3}}$

Step 1.2.18.3

Factor $2$ out of $- 6 x^{2} + 14$ .

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Step 1.2.18.3.1

Factor $2$ out of $- 6 x^{2}$ .

$- \frac{2 (- 3 x^{2}) + 14}{{(x^{2} + 7)}^{3}}$

Step 1.2.18.3.2

Factor $2$ out of $14$ .

$- \frac{2 (- 3 x^{2}) + 2 (7)}{{(x^{2} + 7)}^{3}}$

Step 1.2.18.3.3

Factor $2$ out of $2 (- 3 x^{2}) + 2 (7)$ .

$- \frac{2 (- 3 x^{2} + 7)}{{(x^{2} + 7)}^{3}}$

Step 1.2.18.4

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

$- \frac{2 (- (3 x^{2}) + 7)}{{(x^{2} + 7)}^{3}}$

Step 1.2.18.5

Rewrite $7$ as $- 1 (- 7)$ .

$- \frac{2 (- (3 x^{2}) - 1 (- 7))}{{(x^{2} + 7)}^{3}}$

Step 1.2.18.6

Factor $- 1$ out of $- (3 x^{2}) - 1 (- 7)$ .

$- \frac{2 (- (3 x^{2} - 7))}{{(x^{2} + 7)}^{3}}$

Step 1.2.18.7

Rewrite $- (3 x^{2} - 7)$ as $- 1 (3 x^{2} - 7)$ .

$- \frac{2 (- 1 (3 x^{2} - 7))}{{(x^{2} + 7)}^{3}}$

Step 1.2.18.8

Move the negative in front of the fraction.

$- - \frac{2 (3 x^{2} - 7)}{{(x^{2} + 7)}^{3}}$

Step 1.2.18.9

Multiply $- 1$ by $- 1$ .

$1 \frac{2 (3 x^{2} - 7)}{{(x^{2} + 7)}^{3}}$

Step 1.2.18.10

Multiply $\frac{2 (3 x^{2} - 7)}{{(x^{2} + 7)}^{3}}$ by $1$ .

$f'' (x) = \frac{2 (3 x^{2} - 7)}{{(x^{2} + 7)}^{3}}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{2 (3 x^{2} - 7)}{{(x^{2} + 7)}^{3}}$ .

$\frac{2 (3 x^{2} - 7)}{{(x^{2} + 7)}^{3}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $\frac{2 (3 x^{2} - 7)}{{(x^{2} + 7)}^{3}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$\frac{2 (3 x^{2} - 7)}{{(x^{2} + 7)}^{3}} = 0$

Step 2.2

Set the numerator equal to zero.

$2 (3 x^{2} - 7) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Divide each term in $2 (3 x^{2} - 7) = 0$ by $2$ and simplify.

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Step 2.3.1.1

Divide each term in $2 (3 x^{2} - 7) = 0$ by $2$ .

$\frac{2 (3 x^{2} - 7)}{2} = \frac{0}{2}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $2$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{2 (3 x^{2} - 7)}{2} = \frac{0}{2}$

Step 2.3.1.2.1.2

Divide $3 x^{2} - 7$ by $1$ .

$3 x^{2} - 7 = \frac{0}{2}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $2$ .

$3 x^{2} - 7 = 0$

Step 2.3.2

Add $7$ to both sides of the equation.

$3 x^{2} = 7$

Step 2.3.3

Divide each term in $3 x^{2} = 7$ by $3$ and simplify.

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Step 2.3.3.1

Divide each term in $3 x^{2} = 7$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{7}{3}$

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.

$\frac{3 x^{2}}{3} = \frac{7}{3}$

Step 2.3.3.2.1.2

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

$x^{2} = \frac{7}{3}$

Step 2.3.4

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

$x = \pm \sqrt{\frac{7}{3}}$

Step 2.3.5

Simplify $\pm \sqrt{\frac{7}{3}}$ .

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Step 2.3.5.1

Rewrite $\sqrt{\frac{7}{3}}$ as $\frac{\sqrt{7}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{7}}{\sqrt{3}}$

Step 2.3.5.2

Multiply $\frac{\sqrt{7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{7}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 2.3.5.3

Combine and simplify the denominator.

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Step 2.3.5.3.1

Multiply $\frac{\sqrt{7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{7} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.3.5.3.2

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

$x = \pm \frac{\sqrt{7} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.3.5.3.3

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

$x = \pm \frac{\sqrt{7} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.3.5.3.4

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

$x = \pm \frac{\sqrt{7} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.3.5.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{7} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.3.5.3.6

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

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Step 2.3.5.3.6.1

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

$x = \pm \frac{\sqrt{7} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.3.5.3.6.2

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

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.3.5.3.6.3

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

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.3.5.3.6.4

Cancel the common factor of $2$ .

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Step 2.3.5.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.3.5.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3^{1}}$

Step 2.3.5.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3}$

Step 2.3.5.4

Simplify the numerator.

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Step 2.3.5.4.1

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{7 \cdot 3}}{3}$

Step 2.3.5.4.2

Multiply $7$ by $3$ .

$x = \pm \frac{\sqrt{21}}{3}$

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.

$x = \frac{\sqrt{21}}{3}$

Step 2.3.6.2

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

$x = - \frac{\sqrt{21}}{3}$

Step 2.3.6.3

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

$x = \frac{\sqrt{21}}{3}, - \frac{\sqrt{21}}{3}$

Step 3

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

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Step 3.1

Substitute $\frac{\sqrt{21}}{3}$ in $f (x) = \frac{1}{x^{2} + 7}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $\frac{\sqrt{21}}{3}$ in the expression.

$f (\frac{\sqrt{21}}{3}) = \frac{1}{{(\frac{\sqrt{21}}{3})}^{2} + 7}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify the denominator.

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Step 3.1.2.1.1

Apply the product rule to $\frac{\sqrt{21}}{3}$ .

$f (\frac{\sqrt{21}}{3}) = \frac{1}{\frac{{\sqrt{21}}^{2}}{3^{2}} + 7}$

Step 3.1.2.1.2

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 3.1.2.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$f (\frac{\sqrt{21}}{3}) = \frac{1}{\frac{{(21^{\frac{1}{2}})}^{2}}{3^{2}} + 7}$

Step 3.1.2.1.2.2

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

$f (\frac{\sqrt{21}}{3}) = \frac{1}{\frac{21^{\frac{1}{2} \cdot 2}}{3^{2}} + 7}$

Step 3.1.2.1.2.3

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

$f (\frac{\sqrt{21}}{3}) = \frac{1}{\frac{21^{\frac{2}{2}}}{3^{2}} + 7}$

Step 3.1.2.1.2.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.2.4.1

Cancel the common factor.

$f (\frac{\sqrt{21}}{3}) = \frac{1}{\frac{21^{\frac{2}{2}}}{3^{2}} + 7}$

Step 3.1.2.1.2.4.2

Rewrite the expression.

$f (\frac{\sqrt{21}}{3}) = \frac{1}{\frac{21}{3^{2}} + 7}$

Step 3.1.2.1.2.5

Evaluate the exponent.

$f (\frac{\sqrt{21}}{3}) = \frac{1}{\frac{21}{3^{2}} + 7}$

Step 3.1.2.1.3

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

$f (\frac{\sqrt{21}}{3}) = \frac{1}{\frac{21}{9} + 7}$

Step 3.1.2.1.4

Cancel the common factor of $21$ and $9$ .

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Step 3.1.2.1.4.1

Factor $3$ out of $21$ .

$f (\frac{\sqrt{21}}{3}) = \frac{1}{\frac{3 (7)}{9} + 7}$

Step 3.1.2.1.4.2

Cancel the common factors.

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Step 3.1.2.1.4.2.1

Factor $3$ out of $9$ .

$f (\frac{\sqrt{21}}{3}) = \frac{1}{\frac{3 \cdot 7}{3 \cdot 3} + 7}$

Step 3.1.2.1.4.2.2

Cancel the common factor.

$f (\frac{\sqrt{21}}{3}) = \frac{1}{\frac{3 \cdot 7}{3 \cdot 3} + 7}$

Step 3.1.2.1.4.2.3

Rewrite the expression.

$f (\frac{\sqrt{21}}{3}) = \frac{1}{\frac{7}{3} + 7}$

Step 3.1.2.1.5

To write $7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{\sqrt{21}}{3}) = \frac{1}{\frac{7}{3} + 7 \cdot \frac{3}{3}}$

Step 3.1.2.1.6

Combine $7$ and $\frac{3}{3}$ .

$f (\frac{\sqrt{21}}{3}) = \frac{1}{\frac{7}{3} + \frac{7 \cdot 3}{3}}$

Step 3.1.2.1.7

Combine the numerators over the common denominator.

$f (\frac{\sqrt{21}}{3}) = \frac{1}{\frac{7 + 7 \cdot 3}{3}}$

Step 3.1.2.1.8

Simplify the numerator.

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Step 3.1.2.1.8.1

Multiply $7$ by $3$ .

$f (\frac{\sqrt{21}}{3}) = \frac{1}{\frac{7 + 21}{3}}$

Step 3.1.2.1.8.2

Add $7$ and $21$ .

$f (\frac{\sqrt{21}}{3}) = \frac{1}{\frac{28}{3}}$

Step 3.1.2.2

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{\sqrt{21}}{3}) = 1 (\frac{3}{28})$

Step 3.1.2.3

Multiply $\frac{3}{28}$ by $1$ .

$f (\frac{\sqrt{21}}{3}) = \frac{3}{28}$

Step 3.1.2.4

The final answer is $\frac{3}{28}$ .

$\frac{3}{28}$

Step 3.2

The point found by substituting $\frac{\sqrt{21}}{3}$ in $f (x) = \frac{1}{x^{2} + 7}$ is $(\frac{\sqrt{21}}{3}, \frac{3}{28})$ . This point can be an inflection point.

$(\frac{\sqrt{21}}{3}, \frac{3}{28})$

Step 3.3

Substitute $- \frac{\sqrt{21}}{3}$ in $f (x) = \frac{1}{x^{2} + 7}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $- \frac{\sqrt{21}}{3}$ in the expression.

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{{(- \frac{\sqrt{21}}{3})}^{2} + 7}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify the denominator.

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Step 3.3.2.1.1

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

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Step 3.3.2.1.1.1

Apply the product rule to $- \frac{\sqrt{21}}{3}$ .

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{{(- 1)}^{2} {(\frac{\sqrt{21}}{3})}^{2} + 7}$

Step 3.3.2.1.1.2

Apply the product rule to $\frac{\sqrt{21}}{3}$ .

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{{(- 1)}^{2} (\frac{{\sqrt{21}}^{2}}{3^{2}}) + 7}$

Step 3.3.2.1.2

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

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{1 (\frac{{\sqrt{21}}^{2}}{3^{2}}) + 7}$

Step 3.3.2.1.3

Multiply $\frac{{\sqrt{21}}^{2}}{3^{2}}$ by $1$ .

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{\frac{{\sqrt{21}}^{2}}{3^{2}} + 7}$

Step 3.3.2.1.4

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 3.3.2.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{\frac{{(21^{\frac{1}{2}})}^{2}}{3^{2}} + 7}$

Step 3.3.2.1.4.2

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

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{\frac{21^{\frac{1}{2} \cdot 2}}{3^{2}} + 7}$

Step 3.3.2.1.4.3

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

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{\frac{21^{\frac{2}{2}}}{3^{2}} + 7}$

Step 3.3.2.1.4.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.4.4.1

Cancel the common factor.

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{\frac{21^{\frac{2}{2}}}{3^{2}} + 7}$

Step 3.3.2.1.4.4.2

Rewrite the expression.

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{\frac{21}{3^{2}} + 7}$

Step 3.3.2.1.4.5

Evaluate the exponent.

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{\frac{21}{3^{2}} + 7}$

Step 3.3.2.1.5

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

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{\frac{21}{9} + 7}$

Step 3.3.2.1.6

Cancel the common factor of $21$ and $9$ .

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Step 3.3.2.1.6.1

Factor $3$ out of $21$ .

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{\frac{3 (7)}{9} + 7}$

Step 3.3.2.1.6.2

Cancel the common factors.

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Step 3.3.2.1.6.2.1

Factor $3$ out of $9$ .

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{\frac{3 \cdot 7}{3 \cdot 3} + 7}$

Step 3.3.2.1.6.2.2

Cancel the common factor.

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{\frac{3 \cdot 7}{3 \cdot 3} + 7}$

Step 3.3.2.1.6.2.3

Rewrite the expression.

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{\frac{7}{3} + 7}$

Step 3.3.2.1.7

To write $7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{\frac{7}{3} + 7 \cdot \frac{3}{3}}$

Step 3.3.2.1.8

Combine $7$ and $\frac{3}{3}$ .

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{\frac{7}{3} + \frac{7 \cdot 3}{3}}$

Step 3.3.2.1.9

Combine the numerators over the common denominator.

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{\frac{7 + 7 \cdot 3}{3}}$

Step 3.3.2.1.10

Simplify the numerator.

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Step 3.3.2.1.10.1

Multiply $7$ by $3$ .

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{\frac{7 + 21}{3}}$

Step 3.3.2.1.10.2

Add $7$ and $21$ .

$f (- \frac{\sqrt{21}}{3}) = \frac{1}{\frac{28}{3}}$

Step 3.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{\sqrt{21}}{3}) = 1 (\frac{3}{28})$

Step 3.3.2.3

Multiply $\frac{3}{28}$ by $1$ .

$f (- \frac{\sqrt{21}}{3}) = \frac{3}{28}$

Step 3.3.2.4

The final answer is $\frac{3}{28}$ .

$\frac{3}{28}$

Step 3.4

The point found by substituting $- \frac{\sqrt{21}}{3}$ in $f (x) = \frac{1}{x^{2} + 7}$ is $(- \frac{\sqrt{21}}{3}, \frac{3}{28})$ . This point can be an inflection point.

$(- \frac{\sqrt{21}}{3}, \frac{3}{28})$

Step 3.5

Determine the points that could be inflection points.

$(\frac{\sqrt{21}}{3}, \frac{3}{28}), (- \frac{\sqrt{21}}{3}, \frac{3}{28})$

Step 4

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

$(- \infty, - \frac{\sqrt{21}}{3}) \cup (- \frac{\sqrt{21}}{3}, \frac{\sqrt{21}}{3}) \cup (\frac{\sqrt{21}}{3}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - \frac{\sqrt{21}}{3})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 1.62752523$ in the expression.

$f'' (- 1.62752523) = \frac{2 (3 {(- 1.62752523)}^{2} - 7)}{{({(- 1.62752523)}^{2} + 7)}^{3}}$

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 $- 1.62752523$ to the power of $2$ .

$f'' (- 1.62752523) = \frac{2 (3 \cdot 2.64883837 - 7)}{{({(- 1.62752523)}^{2} + 7)}^{3}}$

Step 5.2.1.2

Multiply $3$ by $2.64883837$ .

$f'' (- 1.62752523) = \frac{2 (7.94651513 - 7)}{{({(- 1.62752523)}^{2} + 7)}^{3}}$

Step 5.2.1.3

Subtract $7$ from $7.94651513$ .

$f'' (- 1.62752523) = \frac{2 \cdot 0.94651513}{{({(- 1.62752523)}^{2} + 7)}^{3}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Raise $- 1.62752523$ to the power of $2$ .

$f'' (- 1.62752523) = \frac{2 \cdot 0.94651513}{{(2.64883837 + 7)}^{3}}$

Step 5.2.2.2

Add $2.64883837$ and $7$ .

$f'' (- 1.62752523) = \frac{2 \cdot 0.94651513}{{9.64883837}^{3}}$

Step 5.2.2.3

Raise $9.64883837$ to the power of $3$ .

$f'' (- 1.62752523) = \frac{2 \cdot 0.94651513}{898.30764509}$

Step 5.2.3

Simplify the expression.

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Step 5.2.3.1

Multiply $2$ by $0.94651513$ .

$f'' (- 1.62752523) = \frac{1.89303027}{898.30764509}$

Step 5.2.3.2

Divide $1.89303027$ by $898.30764509$ .

$f'' (- 1.62752523) = 0.00210732$

Step 5.2.4

The final answer is $0.00210732$ .

$0.00210732$

Step 5.3

At $- 1.62752523$ , the second derivative is $0.00210732$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - \frac{\sqrt{21}}{3})$ .

Increasing on $(- \infty, - \frac{\sqrt{21}}{3})$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(- \frac{\sqrt{21}}{3}, \frac{\sqrt{21}}{3})$ 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.

$f'' (0) = \frac{2 (3 {(0)}^{2} - 7)}{{({(0)}^{2} + 7)}^{3}}$

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 $0$ .

$f'' (0) = \frac{2 (3 \cdot 0 - 7)}{{(0^{2} + 7)}^{3}}$

Step 6.2.1.2

Multiply $3$ by $0$ .

$f'' (0) = \frac{2 (0 - 7)}{{(0^{2} + 7)}^{3}}$

Step 6.2.1.3

Subtract $7$ from $0$ .

$f'' (0) = \frac{2 \cdot - 7}{{(0^{2} + 7)}^{3}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

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

$f'' (0) = \frac{2 \cdot - 7}{{(0 + 7)}^{3}}$

Step 6.2.2.2

Add $0$ and $7$ .

$f'' (0) = \frac{2 \cdot - 7}{7^{3}}$

Step 6.2.2.3

Raise $7$ to the power of $3$ .

$f'' (0) = \frac{2 \cdot - 7}{343}$

Step 6.2.3

Reduce the expression by cancelling the common factors.

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Step 6.2.3.1

Multiply $2$ by $- 7$ .

$f'' (0) = \frac{- 14}{343}$

Step 6.2.3.2

Cancel the common factor of $- 14$ and $343$ .

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Step 6.2.3.2.1

Factor $7$ out of $- 14$ .

$f'' (0) = \frac{7 (- 2)}{343}$

Step 6.2.3.2.2

Cancel the common factors.

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Step 6.2.3.2.2.1

Factor $7$ out of $343$ .

$f'' (0) = \frac{7 \cdot - 2}{7 \cdot 49}$

Step 6.2.3.2.2.2

Cancel the common factor.

$f'' (0) = \frac{7 \cdot - 2}{7 \cdot 49}$

Step 6.2.3.2.2.3

Rewrite the expression.

$f'' (0) = \frac{- 2}{49}$

Step 6.2.3.3

Move the negative in front of the fraction.

$f'' (0) = - \frac{2}{49}$

Step 6.2.4

The final answer is $- \frac{2}{49}$ .

$- \frac{2}{49}$

Step 6.3

At $0$ , the second derivative is $- 0.04081632$ . Since this is negative, the second derivative is decreasing on the interval $(- \frac{\sqrt{21}}{3}, \frac{\sqrt{21}}{3})$

Decreasing on $(- \frac{\sqrt{21}}{3}, \frac{\sqrt{21}}{3})$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(\frac{\sqrt{21}}{3}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $1.62752523$ in the expression.

$f'' (1.62752523) = \frac{2 (3 {(1.62752523)}^{2} - 7)}{{({(1.62752523)}^{2} + 7)}^{3}}$

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 $1.62752523$ to the power of $2$ .

$f'' (1.62752523) = \frac{2 (3 \cdot 2.64883837 - 7)}{{({1.62752523}^{2} + 7)}^{3}}$

Step 7.2.1.2

Multiply $3$ by $2.64883837$ .

$f'' (1.62752523) = \frac{2 (7.94651513 - 7)}{{({1.62752523}^{2} + 7)}^{3}}$

Step 7.2.1.3

Subtract $7$ from $7.94651513$ .

$f'' (1.62752523) = \frac{2 \cdot 0.94651513}{{({1.62752523}^{2} + 7)}^{3}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Raise $1.62752523$ to the power of $2$ .

$f'' (1.62752523) = \frac{2 \cdot 0.94651513}{{(2.64883837 + 7)}^{3}}$

Step 7.2.2.2

Add $2.64883837$ and $7$ .

$f'' (1.62752523) = \frac{2 \cdot 0.94651513}{{9.64883837}^{3}}$

Step 7.2.2.3

Raise $9.64883837$ to the power of $3$ .

$f'' (1.62752523) = \frac{2 \cdot 0.94651513}{898.30764509}$

Step 7.2.3

Simplify the expression.

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Step 7.2.3.1

Multiply $2$ by $0.94651513$ .

$f'' (1.62752523) = \frac{1.89303027}{898.30764509}$

Step 7.2.3.2

Divide $1.89303027$ by $898.30764509$ .

$f'' (1.62752523) = 0.00210732$

Step 7.2.4

The final answer is $0.00210732$ .

$0.00210732$

Step 7.3

At $1.62752523$ , the second derivative is $0.00210732$ . Since this is positive, the second derivative is increasing on the interval $(\frac{\sqrt{21}}{3}, \infty)$ .

Increasing on $(\frac{\sqrt{21}}{3}, \infty)$ 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 $(- \frac{\sqrt{21}}{3}, \frac{3}{28}), (\frac{\sqrt{21}}{3}, \frac{3}{28})$ .

$(- \frac{\sqrt{21}}{3}, \frac{3}{28}), (\frac{\sqrt{21}}{3}, \frac{3}{28})$

Step 9