Calculus Examples

$\frac{e^{x}}{6 + e^{x}}$

Step 1

Write $\frac{e^{x}}{6 + e^{x}}$ as a function.

$f (x) = \frac{e^{x}}{6 + e^{x}}$

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

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 $f (x) = e^{x}$ and $g (x) = 6 + e^{x}$ .

$\frac{(6 + e^{x}) \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [6 + e^{x}]}{{(6 + e^{x})}^{2}}$

Step 2.1.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$\frac{(6 + e^{x}) e^{x} - e^{x} \frac{d}{d x} [6 + e^{x}]}{{(6 + e^{x})}^{2}}$

Step 2.1.1.3

Differentiate.

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Step 2.1.1.3.1

By the Sum Rule, the derivative of $6 + e^{x}$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [e^{x}]$ .

$\frac{(6 + e^{x}) e^{x} - e^{x} (\frac{d}{d x} [6] + \frac{d}{d x} [e^{x}])}{{(6 + e^{x})}^{2}}$

Step 2.1.1.3.2

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$\frac{(6 + e^{x}) e^{x} - e^{x} (0 + \frac{d}{d x} [e^{x}])}{{(6 + e^{x})}^{2}}$

Step 2.1.1.3.3

Add $0$ and $\frac{d}{d x} [e^{x}]$ .

$\frac{(6 + e^{x}) e^{x} - e^{x} \frac{d}{d x} [e^{x}]}{{(6 + e^{x})}^{2}}$

Step 2.1.1.4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$\frac{(6 + e^{x}) e^{x} - e^{x} e^{x}}{{(6 + e^{x})}^{2}}$

Step 2.1.1.5

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

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Step 2.1.1.5.1

Move $e^{x}$ .

$\frac{(6 + e^{x}) e^{x} - (e^{x} e^{x})}{{(6 + e^{x})}^{2}}$

Step 2.1.1.5.2

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

$\frac{(6 + e^{x}) e^{x} - e^{x + x}}{{(6 + e^{x})}^{2}}$

Step 2.1.1.5.3

Add $x$ and $x$ .

$\frac{(6 + e^{x}) e^{x} - e^{2 x}}{{(6 + e^{x})}^{2}}$

Step 2.1.1.6

Simplify.

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Step 2.1.1.6.1

Apply the distributive property.

$\frac{6 e^{x} + e^{x} e^{x} - e^{2 x}}{{(6 + e^{x})}^{2}}$

Step 2.1.1.6.2

Simplify the numerator.

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Step 2.1.1.6.2.1

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

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Step 2.1.1.6.2.1.1

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

$\frac{6 e^{x} + e^{x + x} - e^{2 x}}{{(6 + e^{x})}^{2}}$

Step 2.1.1.6.2.1.2

Add $x$ and $x$ .

$\frac{6 e^{x} + e^{2 x} - e^{2 x}}{{(6 + e^{x})}^{2}}$

Step 2.1.1.6.2.2

Combine the opposite terms in $6 e^{x} + e^{2 x} - e^{2 x}$ .

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Step 2.1.1.6.2.2.1

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

$\frac{6 e^{x} + 0}{{(6 + e^{x})}^{2}}$

Step 2.1.1.6.2.2.2

Add $6 e^{x}$ and $0$ .

$f' (x) = \frac{6 e^{x}}{{(6 + e^{x})}^{2}}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6 e^{x}}{{(6 + e^{x})}^{2}}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{e^{x}}{{(6 + e^{x})}^{2}}]$ .

$6 \frac{d}{d x} [\frac{e^{x}}{{(6 + e^{x})}^{2}}]$

Step 2.1.2.2

$6 \frac{{(6 + e^{x})}^{2} \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [{(6 + e^{x})}^{2}]}{{({(6 + e^{x})}^{2})}^{2}}$

Step 2.1.2.3

Multiply the exponents in ${({(6 + e^{x})}^{2})}^{2}$ .

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Step 2.1.2.3.1

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

$6 \frac{{(6 + e^{x})}^{2} \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [{(6 + e^{x})}^{2}]}{{(6 + e^{x})}^{2 \cdot 2}}$

Step 2.1.2.3.2

Multiply $2$ by $2$ .

$6 \frac{{(6 + e^{x})}^{2} \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [{(6 + e^{x})}^{2}]}{{(6 + e^{x})}^{4}}$

Step 2.1.2.4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$6 \frac{{(6 + e^{x})}^{2} e^{x} - e^{x} \frac{d}{d x} [{(6 + e^{x})}^{2}]}{{(6 + e^{x})}^{4}}$

Step 2.1.2.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 6 + e^{x}$ .

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Step 2.1.2.5.1

To apply the Chain Rule, set $u$ as $6 + e^{x}$ .

$6 \frac{{(6 + e^{x})}^{2} e^{x} - e^{x} (\frac{d}{d u} [u^{2}] \frac{d}{d x} [6 + e^{x}])}{{(6 + e^{x})}^{4}}$

Step 2.1.2.5.2

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

$6 \frac{{(6 + e^{x})}^{2} e^{x} - e^{x} (2 u \frac{d}{d x} [6 + e^{x}])}{{(6 + e^{x})}^{4}}$

Step 2.1.2.5.3

Replace all occurrences of $u$ with $6 + e^{x}$ .

$6 \frac{{(6 + e^{x})}^{2} e^{x} - e^{x} (2 (6 + e^{x}) \frac{d}{d x} [6 + e^{x}])}{{(6 + e^{x})}^{4}}$

Step 2.1.2.6

Differentiate.

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Step 2.1.2.6.1

Multiply $2$ by $- 1$ .

$6 \frac{{(6 + e^{x})}^{2} e^{x} - 2 e^{x} ((6 + e^{x}) \frac{d}{d x} [6 + e^{x}])}{{(6 + e^{x})}^{4}}$

Step 2.1.2.6.2

By the Sum Rule, the derivative of $6 + e^{x}$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [e^{x}]$ .

$6 \frac{{(6 + e^{x})}^{2} e^{x} - 2 e^{x} ((6 + e^{x}) (\frac{d}{d x} [6] + \frac{d}{d x} [e^{x}]))}{{(6 + e^{x})}^{4}}$

Step 2.1.2.6.3

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$6 \frac{{(6 + e^{x})}^{2} e^{x} - 2 e^{x} ((6 + e^{x}) (0 + \frac{d}{d x} [e^{x}]))}{{(6 + e^{x})}^{4}}$

Step 2.1.2.6.4

Add $0$ and $\frac{d}{d x} [e^{x}]$ .

$6 \frac{{(6 + e^{x})}^{2} e^{x} - 2 e^{x} ((6 + e^{x}) \frac{d}{d x} [e^{x}])}{{(6 + e^{x})}^{4}}$

Step 2.1.2.7

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$6 \frac{{(6 + e^{x})}^{2} e^{x} - 2 e^{x} ((6 + e^{x}) e^{x})}{{(6 + e^{x})}^{4}}$

Step 2.1.2.8

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

$6 \frac{{(6 + e^{x})}^{2} e^{x} - 2 e^{x + x} (6 + e^{x})}{{(6 + e^{x})}^{4}}$

Step 2.1.2.9

Add $x$ and $x$ .

$6 \frac{{(6 + e^{x})}^{2} e^{x} - 2 e^{2 x} (6 + e^{x})}{{(6 + e^{x})}^{4}}$

Step 2.1.2.10

Factor $6 + e^{x}$ out of ${(6 + e^{x})}^{2} e^{x} - 2 e^{2 x} (6 + e^{x})$ .

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Step 2.1.2.10.1

Factor $6 + e^{x}$ out of ${(6 + e^{x})}^{2} e^{x}$ .

$6 \frac{(6 + e^{x}) ((6 + e^{x}) e^{x}) - 2 e^{2 x} (6 + e^{x})}{{(6 + e^{x})}^{4}}$

Step 2.1.2.10.2

Factor $6 + e^{x}$ out of $- 2 e^{2 x} (6 + e^{x})$ .

$6 \frac{(6 + e^{x}) ((6 + e^{x}) e^{x}) + (6 + e^{x}) (- 2 e^{2 x})}{{(6 + e^{x})}^{4}}$

Step 2.1.2.10.3

Factor $6 + e^{x}$ out of $(6 + e^{x}) ((6 + e^{x}) e^{x}) + (6 + e^{x}) (- 2 e^{2 x})$ .

$6 \frac{(6 + e^{x}) ((6 + e^{x}) e^{x} - 2 e^{2 x})}{{(6 + e^{x})}^{4}}$

Step 2.1.2.11

Cancel the common factors.

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Step 2.1.2.11.1

Factor $6 + e^{x}$ out of ${(6 + e^{x})}^{4}$ .

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

Step 2.1.2.11.2

Cancel the common factor.

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

Step 2.1.2.11.3

Rewrite the expression.

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

Step 2.1.2.12

Combine $6$ and $\frac{(6 + e^{x}) e^{x} - 2 e^{2 x}}{{(6 + e^{x})}^{3}}$ .

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

Step 2.1.2.13

Simplify.

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Step 2.1.2.13.1

Apply the distributive property.

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

Step 2.1.2.13.2

Apply the distributive property.

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

Step 2.1.2.13.3

Simplify the numerator.

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Step 2.1.2.13.3.1

Simplify each term.

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Step 2.1.2.13.3.1.1

Multiply $6$ by $6$ .

$\frac{36 e^{x} + 6 (e^{x} e^{x}) + 6 (- 2 e^{2 x})}{{(6 + e^{x})}^{3}}$

Step 2.1.2.13.3.1.2

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

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Step 2.1.2.13.3.1.2.1

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

$\frac{36 e^{x} + 6 e^{x + x} + 6 (- 2 e^{2 x})}{{(6 + e^{x})}^{3}}$

Step 2.1.2.13.3.1.2.2

Add $x$ and $x$ .

$\frac{36 e^{x} + 6 e^{2 x} + 6 (- 2 e^{2 x})}{{(6 + e^{x})}^{3}}$

Step 2.1.2.13.3.1.3

Multiply $- 2$ by $6$ .

$\frac{36 e^{x} + 6 e^{2 x} - 12 e^{2 x}}{{(6 + e^{x})}^{3}}$

Step 2.1.2.13.3.2

Subtract $12 e^{2 x}$ from $6 e^{2 x}$ .

$\frac{36 e^{x} - 6 e^{2 x}}{{(6 + e^{x})}^{3}}$

Step 2.1.2.13.4

Simplify the numerator.

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Step 2.1.2.13.4.1

Factor $6$ out of $36 e^{x} - 6 e^{2 x}$ .

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Step 2.1.2.13.4.1.1

Factor $6$ out of $36 e^{x}$ .

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

Step 2.1.2.13.4.1.2

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

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

Step 2.1.2.13.4.1.3

Factor $6$ out of $6 (6 e^{x}) + 6 (- e^{2 x})$ .

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

Step 2.1.2.13.4.2

Rewrite $e^{2 x}$ as ${(e^{x})}^{2}$ .

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

Step 2.1.2.13.4.3

Let $u = e^{x}$ . Substitute $u$ for all occurrences of $e^{x}$ .

$\frac{6 (6 u - u^{2})}{{(6 + e^{x})}^{3}}$

Step 2.1.2.13.4.4

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

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Step 2.1.2.13.4.4.1

Factor $u$ out of $6 u$ .

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

Step 2.1.2.13.4.4.2

Factor $u$ out of $- u^{2}$ .

$\frac{6 (u \cdot 6 + u (- u))}{{(6 + e^{x})}^{3}}$

Step 2.1.2.13.4.4.3

Factor $u$ out of $u \cdot 6 + u (- u)$ .

$\frac{6 (u (6 - u))}{{(6 + e^{x})}^{3}}$

Step 2.1.2.13.4.5

Replace all occurrences of $u$ with $e^{x}$ .

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

Step 2.1.3

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

$\frac{6 e^{x} (6 - e^{x})}{{(6 + e^{x})}^{3}}$

Step 2.2

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

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Step 2.2.1

Set the second derivative equal to $0$ .

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

Step 2.2.2

Set the numerator equal to zero.

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

Step 2.2.3

Solve the equation for $x$ .

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Step 2.2.3.1

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

$e^{x} = 0$

$6 - e^{x} = 0$

Step 2.2.3.2

Set $e^{x}$ equal to $0$ and solve for $x$ .

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Step 2.2.3.2.1

Set $e^{x}$ equal to $0$ .

$e^{x} = 0$

Step 2.2.3.2.2

Solve $e^{x} = 0$ for $x$ .

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Step 2.2.3.2.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x}) = \ln (0)$

Step 2.2.3.2.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 2.2.3.2.2.3

There is no solution for $e^{x} = 0$

No solution

Step 2.2.3.3

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

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Step 2.2.3.3.1

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

$6 - e^{x} = 0$

Step 2.2.3.3.2

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

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Step 2.2.3.3.2.1

Subtract $6$ from both sides of the equation.

$- e^{x} = - 6$

Step 2.2.3.3.2.2

Divide each term in $- e^{x} = - 6$ by $- 1$ and simplify.

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Step 2.2.3.3.2.2.1

Divide each term in $- e^{x} = - 6$ by $- 1$ .

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

Step 2.2.3.3.2.2.2

Simplify the left side.

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Step 2.2.3.3.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.3.3.2.2.2.2

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

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

Step 2.2.3.3.2.2.3

Simplify the right side.

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Step 2.2.3.3.2.2.3.1

Divide $- 6$ by $- 1$ .

$e^{x} = 6$

Step 2.2.3.3.2.3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x}) = \ln (6)$

Step 2.2.3.3.2.4

Expand the left side.

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Step 2.2.3.3.2.4.1

Expand $\ln (e^{x})$ by moving $x$ outside the logarithm.

$x \ln (e) = \ln (6)$

Step 2.2.3.3.2.4.2

The natural logarithm of $e$ is $1$ .

$x \cdot 1 = \ln (6)$

Step 2.2.3.3.2.4.3

Multiply $x$ by $1$ .

$x = \ln (6)$

Step 2.2.3.4

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

$x = \ln (6)$

Step 3

Find the domain of $f (x) = \frac{e^{x}}{6 + e^{x}}$ .

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Step 3.1

Set the denominator in $\frac{e^{x}}{6 + e^{x}}$ equal to $0$ to find where the expression is undefined.

$6 + e^{x} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Subtract $6$ from both sides of the equation.

$e^{x} = - 6$

Step 3.2.2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x}) = \ln (- 6)$

Step 3.2.3

The equation cannot be solved because $\ln (- 6)$ is undefined.

Undefined

Step 3.2.4

There is no solution for $e^{x} = - 6$

No solution

Step 3.3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

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

Step 5

Substitute any number from the interval $(- \infty, \ln (6))$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

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

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

Anything raised to $0$ is $1$ .

$f'' (0) = \frac{6 e^{0} (6 - 1 \cdot 1)}{{(6 + e^{0})}^{3}}$

Step 5.2.1.2

Multiply $- 1$ by $1$ .

$f'' (0) = \frac{6 e^{0} (6 - 1)}{{(6 + e^{0})}^{3}}$

Step 5.2.1.3

Subtract $1$ from $6$ .

$f'' (0) = \frac{6 e^{0} \cdot 5}{{(6 + e^{0})}^{3}}$

Step 5.2.1.4

Multiply $5$ by $6$ .

$f'' (0) = \frac{30 e^{0}}{{(6 + e^{0})}^{3}}$

Step 5.2.1.5

Anything raised to $0$ is $1$ .

$f'' (0) = \frac{30 \cdot 1}{{(6 + e^{0})}^{3}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Anything raised to $0$ is $1$ .

$f'' (0) = \frac{30 \cdot 1}{{(6 + 1)}^{3}}$

Step 5.2.2.2

Add $6$ and $1$ .

$f'' (0) = \frac{30 \cdot 1}{7^{3}}$

Step 5.2.2.3

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

$f'' (0) = \frac{30 \cdot 1}{343}$

Step 5.2.3

Multiply $30$ by $1$ .

$f'' (0) = \frac{30}{343}$

Step 5.2.4

The final answer is $\frac{30}{343}$ .

$\frac{30}{343}$

Step 5.3

The graph is concave up on the interval $(- \infty, \ln (6))$ because $f'' (0)$ is positive.

Concave up on $(- \infty, \ln (6))$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(\ln (6), \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $4$ in the expression.

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

Step 6.2

The final answer is $\frac{6 e^{4} (6 - e^{4})}{{(6 + e^{4})}^{3}}$ .

$\frac{6 e^{4} (6 - e^{4})}{{(6 + e^{4})}^{3}}$

Step 6.3

The graph is concave down on the interval $(\ln (6), \infty)$ because $f'' (4)$ is negative.

Concave down on $(\ln (6), \infty)$ since $f'' (x)$ is negative

Step 7

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave up on $(- \infty, \ln (6))$ since $f'' (x)$ is positive

Concave down on $(\ln (6), \infty)$ since $f'' (x)$ is negative

Step 8