Calculus Examples

$f (x) = 20 x^{\frac{1}{2}} - \frac{1}{2} x^{20000}$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $20 x^{\frac{1}{2}} - \frac{1}{2} x^{20000}$ with respect to $x$ is $\frac{d}{d x} [20 x^{\frac{1}{2}}] + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$ .

$\frac{d}{d x} [20 x^{\frac{1}{2}}] + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 1.2

Evaluate $\frac{d}{d x} [20 x^{\frac{1}{2}}]$ .

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Step 1.2.1

Since $20$ is constant with respect to $x$ , the derivative of $20 x^{\frac{1}{2}}$ with respect to $x$ is $20 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$20 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 1.2.2

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

$20 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$20 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 1.2.4

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

$20 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 1.2.5

Combine the numerators over the common denominator.

$20 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $2$ .

$20 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 1.2.6.2

Subtract $2$ from $1$ .

$20 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 1.2.7

Move the negative in front of the fraction.

$20 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 1.2.8

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$20 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 1.2.9

Combine $20$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{20 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 1.2.10

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{20}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 1.2.11

Factor $2$ out of $20$ .

$\frac{2 \cdot 10}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 1.2.12

Cancel the common factors.

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Step 1.2.12.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$\frac{2 \cdot 10}{2 (x^{\frac{1}{2}})} + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 1.2.12.2

Cancel the common factor.

$\frac{2 \cdot 10}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 1.2.12.3

Rewrite the expression.

$\frac{10}{x^{\frac{1}{2}}} + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 1.3

Evaluate $\frac{d}{d x} [- \frac{1}{2} x^{20000}]$ .

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Step 1.3.1

Since $- \frac{1}{2}$ is constant with respect to $x$ , the derivative of $- \frac{1}{2} x^{20000}$ with respect to $x$ is $- \frac{1}{2} \frac{d}{d x} [x^{20000}]$ .

$\frac{10}{x^{\frac{1}{2}}} - \frac{1}{2} \frac{d}{d x} [x^{20000}]$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 20000$ .

$\frac{10}{x^{\frac{1}{2}}} - \frac{1}{2} (20000 x^{19999})$

Step 1.3.3

Multiply $20000$ by $- 1$ .

$\frac{10}{x^{\frac{1}{2}}} - 20000 (\frac{1}{2}) x^{19999}$

Step 1.3.4

Combine $- 20000$ and $\frac{1}{2}$ .

$\frac{10}{x^{\frac{1}{2}}} + \frac{- 20000}{2} x^{19999}$

Step 1.3.5

Combine $\frac{- 20000}{2}$ and $x^{19999}$ .

$\frac{10}{x^{\frac{1}{2}}} + \frac{- 20000 x^{19999}}{2}$

Step 1.3.6

Cancel the common factor of $- 20000$ and $2$ .

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Step 1.3.6.1

Factor $2$ out of $- 20000 x^{19999}$ .

$\frac{10}{x^{\frac{1}{2}}} + \frac{2 (- 10000 x^{19999})}{2}$

Step 1.3.6.2

Cancel the common factors.

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Step 1.3.6.2.1

Factor $2$ out of $2$ .

$\frac{10}{x^{\frac{1}{2}}} + \frac{2 (- 10000 x^{19999})}{2 (1)}$

Step 1.3.6.2.2

Cancel the common factor.

$\frac{10}{x^{\frac{1}{2}}} + \frac{2 (- 10000 x^{19999})}{2 \cdot 1}$

Step 1.3.6.2.3

Rewrite the expression.

$\frac{10}{x^{\frac{1}{2}}} + \frac{- 10000 x^{19999}}{1}$

Step 1.3.6.2.4

Divide $- 10000 x^{19999}$ by $1$ .

$\frac{10}{x^{\frac{1}{2}}} - 10000 x^{19999}$

Step 1.4

Reorder terms.

$- 10000 x^{19999} + \frac{10}{x^{\frac{1}{2}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- 10000 x^{19999} + \frac{10}{x^{\frac{1}{2}}}$ with respect to $x$ is $\frac{d}{d x} [- 10000 x^{19999}] + \frac{d}{d x} [\frac{10}{x^{\frac{1}{2}}}]$ .

$f'' (x) = \frac{d}{d x} (- 10000 x^{19999}) + \frac{d}{d x} (\frac{10}{x^{\frac{1}{2}}})$

Step 2.2

Evaluate $\frac{d}{d x} [- 10000 x^{19999}]$ .

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Step 2.2.1

Since $- 10000$ is constant with respect to $x$ , the derivative of $- 10000 x^{19999}$ with respect to $x$ is $- 10000 \frac{d}{d x} [x^{19999}]$ .

$f'' (x) = - 10000 \frac{d}{d x} x^{19999} + \frac{d}{d x} (\frac{10}{x^{\frac{1}{2}}})$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 19999$ .

$f'' (x) = - 10000 (19999 x^{19998}) + \frac{d}{d x} (\frac{10}{x^{\frac{1}{2}}})$

Step 2.2.3

Multiply $19999$ by $- 10000$ .

$f'' (x) = - 199990000 x^{19998} + \frac{d}{d x} (\frac{10}{x^{\frac{1}{2}}})$

Step 2.3

Evaluate $\frac{d}{d x} [\frac{10}{x^{\frac{1}{2}}}]$ .

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Step 2.3.1

Since $10$ is constant with respect to $x$ , the derivative of $\frac{10}{x^{\frac{1}{2}}}$ with respect to $x$ is $10 \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$ .

$f'' (x) = - 199990000 x^{19998} + 10 \frac{d}{d x} (\frac{1}{x^{\frac{1}{2}}})$

Step 2.3.2

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

$f'' (x) = - 199990000 x^{19998} + 10 \frac{d}{d x} ({(x^{\frac{1}{2}})}^{- 1})$

Step 2.3.3

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 $g (x) = x^{\frac{1}{2}}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{2}}$ .

$f'' (x) = - 199990000 x^{19998} + 10 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{\frac{1}{2}}))$

Step 2.3.3.2

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

$f'' (x) = - 199990000 x^{19998} + 10 (- u^{- 2} \frac{d}{d x} x^{\frac{1}{2}})$

Step 2.3.3.3

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

$f'' (x) = - 199990000 x^{19998} + 10 (- {(x^{\frac{1}{2}})}^{- 2} \frac{d}{d x} x^{\frac{1}{2}})$

Step 2.3.4

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

$f'' (x) = - 199990000 x^{19998} + 10 (- {(x^{\frac{1}{2}})}^{- 2} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.3.5

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 2}$ .

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Step 2.3.5.1

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

$f'' (x) = - 199990000 x^{19998} + 10 (- x^{\frac{1}{2} \cdot - 2} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.3.5.2

Cancel the common factor of $2$ .

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Step 2.3.5.2.1

Factor $2$ out of $- 2$ .

$f'' (x) = - 199990000 x^{19998} + 10 (- x^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.3.5.2.2

Cancel the common factor.

$f'' (x) = - 199990000 x^{19998} + 10 (- x^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.3.5.2.3

Rewrite the expression.

$f'' (x) = - 199990000 x^{19998} + 10 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.3.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = - 199990000 x^{19998} + 10 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))$

Step 2.3.7

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

$f'' (x) = - 199990000 x^{19998} + 10 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 2.3.8

Combine the numerators over the common denominator.

$f'' (x) = - 199990000 x^{19998} + 10 (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}))$

Step 2.3.9

Simplify the numerator.

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Step 2.3.9.1

Multiply $- 1$ by $2$ .

$f'' (x) = - 199990000 x^{19998} + 10 (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 2}{2}}))$

Step 2.3.9.2

Subtract $2$ from $1$ .

$f'' (x) = - 199990000 x^{19998} + 10 (- x^{- 1} (\frac{1}{2} x^{\frac{- 1}{2}}))$

Step 2.3.10

Move the negative in front of the fraction.

$f'' (x) = - 199990000 x^{19998} + 10 (- x^{- 1} (\frac{1}{2} x^{- \frac{1}{2}}))$

Step 2.3.11

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$f'' (x) = - 199990000 x^{19998} + 10 (- x^{- 1} \frac{x^{- \frac{1}{2}}}{2})$

Step 2.3.12

Combine $\frac{x^{- \frac{1}{2}}}{2}$ and $x^{- 1}$ .

$f'' (x) = - 199990000 x^{19998} + 10 (- \frac{x^{- \frac{1}{2}} x^{- 1}}{2})$

Step 2.3.13

Multiply $x^{- \frac{1}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 2.3.13.1

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

$f'' (x) = - 199990000 x^{19998} + 10 (- \frac{x^{- \frac{1}{2} - 1}}{2})$

Step 2.3.13.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = - 199990000 x^{19998} + 10 (- \frac{x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}}{2})$

Step 2.3.13.3

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

$f'' (x) = - 199990000 x^{19998} + 10 (- \frac{x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2})$

Step 2.3.13.4

Combine the numerators over the common denominator.

$f'' (x) = - 199990000 x^{19998} + 10 (- \frac{x^{\frac{- 1 - 1 \cdot 2}{2}}}{2})$

Step 2.3.13.5

Simplify the numerator.

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Step 2.3.13.5.1

Multiply $- 1$ by $2$ .

$f'' (x) = - 199990000 x^{19998} + 10 (- \frac{x^{\frac{- 1 - 2}{2}}}{2})$

Step 2.3.13.5.2

Subtract $2$ from $- 1$ .

$f'' (x) = - 199990000 x^{19998} + 10 (- \frac{x^{\frac{- 3}{2}}}{2})$

Step 2.3.13.6

Move the negative in front of the fraction.

$f'' (x) = - 199990000 x^{19998} + 10 (- \frac{x^{- \frac{3}{2}}}{2})$

Step 2.3.14

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - 199990000 x^{19998} + 10 (- \frac{1}{2 x^{\frac{3}{2}}})$

Step 2.3.15

Multiply $- 1$ by $10$ .

$f'' (x) = - 199990000 x^{19998} - 10 \frac{1}{2 x^{\frac{3}{2}}}$

Step 2.3.16

Combine $- 10$ and $\frac{1}{2 x^{\frac{3}{2}}}$ .

$f'' (x) = - 199990000 x^{19998} + \frac{- 10}{2 x^{\frac{3}{2}}}$

Step 2.3.17

Factor $2$ out of $- 10$ .

$f'' (x) = - 199990000 x^{19998} + \frac{2 \cdot - 5}{2 x^{\frac{3}{2}}}$

Step 2.3.18

Cancel the common factors.

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Step 2.3.18.1

Factor $2$ out of $2 x^{\frac{3}{2}}$ .

$f'' (x) = - 199990000 x^{19998} + \frac{2 \cdot - 5}{2 (x^{\frac{3}{2}})}$

Step 2.3.18.2

Cancel the common factor.

$f'' (x) = - 199990000 x^{19998} + \frac{2 \cdot - 5}{2 x^{\frac{3}{2}}}$

Step 2.3.18.3

Rewrite the expression.

$f'' (x) = - 199990000 x^{19998} + \frac{- 5}{x^{\frac{3}{2}}}$

Step 2.3.19

Move the negative in front of the fraction.

$f'' (x) = - 199990000 x^{19998} - \frac{5}{x^{\frac{3}{2}}}$

Step 3

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

$- 10000 x^{19999} + \frac{10}{x^{\frac{1}{2}}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

By the Sum Rule, the derivative of $20 x^{\frac{1}{2}} - \frac{1}{2} x^{20000}$ with respect to $x$ is $\frac{d}{d x} [20 x^{\frac{1}{2}}] + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$ .

$\frac{d}{d x} [20 x^{\frac{1}{2}}] + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 4.1.2

Evaluate $\frac{d}{d x} [20 x^{\frac{1}{2}}]$ .

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Step 4.1.2.1

Since $20$ is constant with respect to $x$ , the derivative of $20 x^{\frac{1}{2}}$ with respect to $x$ is $20 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$20 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 4.1.2.2

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

$20 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 4.1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$20 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 4.1.2.4

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

$20 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 4.1.2.5

Combine the numerators over the common denominator.

$20 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 4.1.2.6

Simplify the numerator.

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Step 4.1.2.6.1

Multiply $- 1$ by $2$ .

$20 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 4.1.2.6.2

Subtract $2$ from $1$ .

$20 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 4.1.2.7

Move the negative in front of the fraction.

$20 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 4.1.2.8

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$20 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 4.1.2.9

Combine $20$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{20 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 4.1.2.10

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{20}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 4.1.2.11

Factor $2$ out of $20$ .

$\frac{2 \cdot 10}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 4.1.2.12

Cancel the common factors.

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Step 4.1.2.12.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$\frac{2 \cdot 10}{2 (x^{\frac{1}{2}})} + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 4.1.2.12.2

Cancel the common factor.

$\frac{2 \cdot 10}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 4.1.2.12.3

Rewrite the expression.

$\frac{10}{x^{\frac{1}{2}}} + \frac{d}{d x} [- \frac{1}{2} x^{20000}]$

Step 4.1.3

Evaluate $\frac{d}{d x} [- \frac{1}{2} x^{20000}]$ .

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Step 4.1.3.1

Since $- \frac{1}{2}$ is constant with respect to $x$ , the derivative of $- \frac{1}{2} x^{20000}$ with respect to $x$ is $- \frac{1}{2} \frac{d}{d x} [x^{20000}]$ .

$\frac{10}{x^{\frac{1}{2}}} - \frac{1}{2} \frac{d}{d x} [x^{20000}]$

Step 4.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 20000$ .

$\frac{10}{x^{\frac{1}{2}}} - \frac{1}{2} (20000 x^{19999})$

Step 4.1.3.3

Multiply $20000$ by $- 1$ .

$\frac{10}{x^{\frac{1}{2}}} - 20000 (\frac{1}{2}) x^{19999}$

Step 4.1.3.4

Combine $- 20000$ and $\frac{1}{2}$ .

$\frac{10}{x^{\frac{1}{2}}} + \frac{- 20000}{2} x^{19999}$

Step 4.1.3.5

Combine $\frac{- 20000}{2}$ and $x^{19999}$ .

$\frac{10}{x^{\frac{1}{2}}} + \frac{- 20000 x^{19999}}{2}$

Step 4.1.3.6

Cancel the common factor of $- 20000$ and $2$ .

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Step 4.1.3.6.1

Factor $2$ out of $- 20000 x^{19999}$ .

$\frac{10}{x^{\frac{1}{2}}} + \frac{2 (- 10000 x^{19999})}{2}$

Step 4.1.3.6.2

Cancel the common factors.

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Step 4.1.3.6.2.1

Factor $2$ out of $2$ .

$\frac{10}{x^{\frac{1}{2}}} + \frac{2 (- 10000 x^{19999})}{2 (1)}$

Step 4.1.3.6.2.2

Cancel the common factor.

$\frac{10}{x^{\frac{1}{2}}} + \frac{2 (- 10000 x^{19999})}{2 \cdot 1}$

Step 4.1.3.6.2.3

Rewrite the expression.

$\frac{10}{x^{\frac{1}{2}}} + \frac{- 10000 x^{19999}}{1}$

Step 4.1.3.6.2.4

Divide $- 10000 x^{19999}$ by $1$ .

$\frac{10}{x^{\frac{1}{2}}} - 10000 x^{19999}$

Step 4.1.4

Reorder terms.

$f' (x) = - 10000 x^{19999} + \frac{10}{x^{\frac{1}{2}}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- 10000 x^{19999} + \frac{10}{x^{\frac{1}{2}}}$ .

$- 10000 x^{19999} + \frac{10}{x^{\frac{1}{2}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $- 10000 x^{19999} + \frac{10}{x^{\frac{1}{2}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- 10000 x^{19999} + \frac{10}{x^{\frac{1}{2}}} = 0$

Step 5.2

Find the LCD of the terms in the equation.

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Step 5.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x^{\frac{1}{2}}, 1$

Step 5.2.2

The LCM of one and any expression is the expression.

$x^{\frac{1}{2}}$

Step 5.3

Multiply each term in $- 10000 x^{19999} + \frac{10}{x^{\frac{1}{2}}} = 0$ by $x^{\frac{1}{2}}$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in $- 10000 x^{19999} + \frac{10}{x^{\frac{1}{2}}} = 0$ by $x^{\frac{1}{2}}$ .

$- 10000 x^{19999} x^{\frac{1}{2}} + \frac{10}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Simplify each term.

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Step 5.3.2.1.1

Multiply $x^{19999}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 5.3.2.1.1.1

Move $x^{\frac{1}{2}}$ .

$- 10000 (x^{\frac{1}{2}} x^{19999}) + \frac{10}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 5.3.2.1.1.2

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

$- 10000 x^{\frac{1}{2} + 19999} + \frac{10}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 5.3.2.1.1.3

To write $19999$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 10000 x^{\frac{1}{2} + 19999 \cdot \frac{2}{2}} + \frac{10}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 5.3.2.1.1.4

Combine $19999$ and $\frac{2}{2}$ .

$- 10000 x^{\frac{1}{2} + \frac{19999 \cdot 2}{2}} + \frac{10}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 5.3.2.1.1.5

Combine the numerators over the common denominator.

$- 10000 x^{\frac{1 + 19999 \cdot 2}{2}} + \frac{10}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 5.3.2.1.1.6

Simplify the numerator.

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Step 5.3.2.1.1.6.1

Multiply $19999$ by $2$ .

$- 10000 x^{\frac{1 + 39998}{2}} + \frac{10}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 5.3.2.1.1.6.2

Add $1$ and $39998$ .

$- 10000 x^{\frac{39999}{2}} + \frac{10}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 5.3.2.1.2

Cancel the common factor of $x^{\frac{1}{2}}$ .

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Step 5.3.2.1.2.1

Cancel the common factor.

$- 10000 x^{\frac{39999}{2}} + \frac{10}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 5.3.2.1.2.2

Rewrite the expression.

$- 10000 x^{\frac{39999}{2}} + 10 = 0 x^{\frac{1}{2}}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Multiply $0$ by $x^{\frac{1}{2}}$ .

$- 10000 x^{\frac{39999}{2}} + 10 = 0$

Step 5.4

Solve the equation.

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Step 5.4.1

Subtract $10$ from both sides of the equation.

$- 10000 x^{\frac{39999}{2}} = - 10$

Step 5.4.2

Raise each side of the equation to the power of $\frac{2}{39999}$ to eliminate the fractional exponent on the left side.

${(- 10000 x^{\frac{39999}{2}})}^{\frac{2}{39999}} = {(- 10)}^{\frac{2}{39999}}$

Step 5.4.3

Simplify the left side.

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Step 5.4.3.1

Simplify ${(- 10000 x^{\frac{39999}{2}})}^{\frac{2}{39999}}$ .

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Step 5.4.3.1.1

Apply the product rule to $- 10000 x^{\frac{39999}{2}}$ .

${(- 10000)}^{\frac{2}{39999}} {(x^{\frac{39999}{2}})}^{\frac{2}{39999}} = {(- 10)}^{\frac{2}{39999}}$

Step 5.4.3.1.2

Multiply the exponents in ${(x^{\frac{39999}{2}})}^{\frac{2}{39999}}$ .

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Step 5.4.3.1.2.1

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

${(- 10000)}^{\frac{2}{39999}} x^{\frac{39999}{2} \cdot \frac{2}{39999}} = {(- 10)}^{\frac{2}{39999}}$

Step 5.4.3.1.2.2

Cancel the common factor of $39999$ .

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Step 5.4.3.1.2.2.1

Cancel the common factor.

${(- 10000)}^{\frac{2}{39999}} x^{\frac{39999}{2} \cdot \frac{2}{39999}} = {(- 10)}^{\frac{2}{39999}}$

Step 5.4.3.1.2.2.2

Rewrite the expression.

${(- 10000)}^{\frac{2}{39999}} x^{\frac{1}{2} \cdot 2} = {(- 10)}^{\frac{2}{39999}}$

Step 5.4.3.1.2.3

Cancel the common factor of $2$ .

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Step 5.4.3.1.2.3.1

Cancel the common factor.

${(- 10000)}^{\frac{2}{39999}} x^{\frac{1}{2} \cdot 2} = {(- 10)}^{\frac{2}{39999}}$

Step 5.4.3.1.2.3.2

Rewrite the expression.

${(- 10000)}^{\frac{2}{39999}} x^{1} = {(- 10)}^{\frac{2}{39999}}$

Step 5.4.3.1.3

Simplify.

${(- 10000)}^{\frac{2}{39999}} x = {(- 10)}^{\frac{2}{39999}}$

Step 5.4.3.1.4

Reorder factors in ${(- 10000)}^{\frac{2}{39999}} x$ .

$x {(- 10000)}^{\frac{2}{39999}} = {(- 10)}^{\frac{2}{39999}}$

Step 5.4.4

Divide each term in $x {(- 10000)}^{\frac{2}{39999}} = {(- 10)}^{\frac{2}{39999}}$ by ${(- 10000)}^{\frac{2}{39999}}$ and simplify.

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Step 5.4.4.1

Divide each term in $x {(- 10000)}^{\frac{2}{39999}} = {(- 10)}^{\frac{2}{39999}}$ by ${(- 10000)}^{\frac{2}{39999}}$ .

$\frac{x {(- 10000)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}} = \frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}$

Step 5.4.4.2

Simplify the left side.

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Step 5.4.4.2.1

Cancel the common factor.

$\frac{x {(- 10000)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}} = \frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}$

Step 5.4.4.2.2

Divide $x$ by $1$ .

$x = \frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$- 10000 x^{19999} + \frac{10}{\sqrt{x^{1}}}$

Step 6.1.2

Anything raised to $1$ is the base itself.

$- 10000 x^{19999} + \frac{10}{\sqrt{x}}$

Step 6.2

Set the denominator in $\frac{10}{\sqrt{x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x} = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x}}^{2} = 0^{2}$

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

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

${(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.3.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{2}$

Step 6.3.2.2.1.2

Simplify.

$x = 0^{2}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

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

$x = 0$

Step 6.4

Set the radicand in $\sqrt{x}$ less than $0$ to find where the expression is undefined.

$x < 0$

Step 6.5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 7

Critical points to evaluate.

$x = \frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}, 0$

Step 8

Evaluate the second derivative at $x = \frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 199990000 {(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{19998} - \frac{5}{{(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{\frac{3}{2}}}$

Step 9

Simplify each term.

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Step 9.1

Apply the product rule to $\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}$ .

$- 199990000 \frac{{({(- 10)}^{\frac{2}{39999}})}^{19998}}{{({(- 10000)}^{\frac{2}{39999}})}^{19998}} - \frac{5}{{(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{\frac{3}{2}}}$

Step 9.2

Multiply the exponents in ${({(- 10)}^{\frac{2}{39999}})}^{19998}$ .

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Step 9.2.1

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

$- 199990000 \frac{{(- 10)}^{\frac{2}{39999} \cdot 19998}}{{({(- 10000)}^{\frac{2}{39999}})}^{19998}} - \frac{5}{{(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{\frac{3}{2}}}$

Step 9.2.2

Cancel the common factor of $3$ .

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Step 9.2.2.1

Factor $3$ out of $39999$ .

$- 199990000 \frac{{(- 10)}^{\frac{2}{3 (13333)} \cdot 19998}}{{({(- 10000)}^{\frac{2}{39999}})}^{19998}} - \frac{5}{{(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{\frac{3}{2}}}$

Step 9.2.2.2

Factor $3$ out of $19998$ .

$- 199990000 \frac{{(- 10)}^{\frac{2}{3 \cdot 13333} \cdot (3 \cdot 6666)}}{{({(- 10000)}^{\frac{2}{39999}})}^{19998}} - \frac{5}{{(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{\frac{3}{2}}}$

Step 9.2.2.3

Cancel the common factor.

Step 9.2.2.4

Rewrite the expression.

$- 199990000 \frac{{(- 10)}^{\frac{2}{13333} \cdot 6666}}{{({(- 10000)}^{\frac{2}{39999}})}^{19998}} - \frac{5}{{(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{\frac{3}{2}}}$

Step 9.2.3

Combine $\frac{2}{13333}$ and $6666$ .

$- 199990000 \frac{{(- 10)}^{\frac{2 \cdot 6666}{13333}}}{{({(- 10000)}^{\frac{2}{39999}})}^{19998}} - \frac{5}{{(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{\frac{3}{2}}}$

Step 9.2.4

Multiply $2$ by $6666$ .

$- 199990000 \frac{{(- 10)}^{\frac{13332}{13333}}}{{({(- 10000)}^{\frac{2}{39999}})}^{19998}} - \frac{5}{{(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{\frac{3}{2}}}$

Step 9.3

Multiply the exponents in ${({(- 10000)}^{\frac{2}{39999}})}^{19998}$ .

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Step 9.3.1

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

$- 199990000 \frac{{(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{2}{39999} \cdot 19998}} - \frac{5}{{(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{\frac{3}{2}}}$

Step 9.3.2

Cancel the common factor of $3$ .

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Step 9.3.2.1

Factor $3$ out of $39999$ .

$- 199990000 \frac{{(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{2}{3 (13333)} \cdot 19998}} - \frac{5}{{(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{\frac{3}{2}}}$

Step 9.3.2.2

Factor $3$ out of $19998$ .

$- 199990000 \frac{{(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{2}{3 \cdot 13333} \cdot (3 \cdot 6666)}} - \frac{5}{{(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{\frac{3}{2}}}$

Step 9.3.2.3

Cancel the common factor.

Step 9.3.2.4

Rewrite the expression.

$- 199990000 \frac{{(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{2}{13333} \cdot 6666}} - \frac{5}{{(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{\frac{3}{2}}}$

Step 9.3.3

Combine $\frac{2}{13333}$ and $6666$ .

$- 199990000 \frac{{(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{2 \cdot 6666}{13333}}} - \frac{5}{{(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{\frac{3}{2}}}$

Step 9.3.4

Multiply $2$ by $6666$ .

$- 199990000 \frac{{(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - \frac{5}{{(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{\frac{3}{2}}}$

Step 9.4

Combine $- 199990000$ and $\frac{{(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}}$ .

$\frac{- 199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - \frac{5}{{(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{\frac{3}{2}}}$

Step 9.5

Move the negative in front of the fraction.

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - \frac{5}{{(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{\frac{3}{2}}}$

Step 9.6

Simplify the denominator.

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Step 9.6.1

Apply the product rule to $\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}$ .

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - \frac{5}{\frac{{({(- 10)}^{\frac{2}{39999}})}^{\frac{3}{2}}}{{({(- 10000)}^{\frac{2}{39999}})}^{\frac{3}{2}}}}$

Step 9.6.2

Multiply the exponents in ${({(- 10)}^{\frac{2}{39999}})}^{\frac{3}{2}}$ .

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Step 9.6.2.1

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

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - \frac{5}{\frac{{(- 10)}^{\frac{2}{39999} \cdot \frac{3}{2}}}{{({(- 10000)}^{\frac{2}{39999}})}^{\frac{3}{2}}}}$

Step 9.6.2.2

Cancel the common factor of $2$ .

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Step 9.6.2.2.1

Cancel the common factor.

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - \frac{5}{\frac{{(- 10)}^{\frac{2}{39999} \cdot \frac{3}{2}}}{{({(- 10000)}^{\frac{2}{39999}})}^{\frac{3}{2}}}}$

Step 9.6.2.2.2

Rewrite the expression.

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - \frac{5}{\frac{{(- 10)}^{\frac{1}{39999} \cdot 3}}{{({(- 10000)}^{\frac{2}{39999}})}^{\frac{3}{2}}}}$

Step 9.6.2.3

Cancel the common factor of $3$ .

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Step 9.6.2.3.1

Factor $3$ out of $39999$ .

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - \frac{5}{\frac{{(- 10)}^{\frac{1}{3 (13333)} \cdot 3}}{{({(- 10000)}^{\frac{2}{39999}})}^{\frac{3}{2}}}}$

Step 9.6.2.3.2

Cancel the common factor.

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - \frac{5}{\frac{{(- 10)}^{\frac{1}{3 \cdot 13333} \cdot 3}}{{({(- 10000)}^{\frac{2}{39999}})}^{\frac{3}{2}}}}$

Step 9.6.2.3.3

Rewrite the expression.

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - \frac{5}{\frac{{(- 10)}^{\frac{1}{13333}}}{{({(- 10000)}^{\frac{2}{39999}})}^{\frac{3}{2}}}}$

Step 9.6.3

Multiply the exponents in ${({(- 10000)}^{\frac{2}{39999}})}^{\frac{3}{2}}$ .

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Step 9.6.3.1

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

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - \frac{5}{\frac{{(- 10)}^{\frac{1}{13333}}}{{(- 10000)}^{\frac{2}{39999} \cdot \frac{3}{2}}}}$

Step 9.6.3.2

Cancel the common factor of $2$ .

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Step 9.6.3.2.1

Cancel the common factor.

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - \frac{5}{\frac{{(- 10)}^{\frac{1}{13333}}}{{(- 10000)}^{\frac{2}{39999} \cdot \frac{3}{2}}}}$

Step 9.6.3.2.2

Rewrite the expression.

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - \frac{5}{\frac{{(- 10)}^{\frac{1}{13333}}}{{(- 10000)}^{\frac{1}{39999} \cdot 3}}}$

Step 9.6.3.3

Cancel the common factor of $3$ .

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Step 9.6.3.3.1

Factor $3$ out of $39999$ .

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - \frac{5}{\frac{{(- 10)}^{\frac{1}{13333}}}{{(- 10000)}^{\frac{1}{3 (13333)} \cdot 3}}}$

Step 9.6.3.3.2

Cancel the common factor.

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - \frac{5}{\frac{{(- 10)}^{\frac{1}{13333}}}{{(- 10000)}^{\frac{1}{3 \cdot 13333} \cdot 3}}}$

Step 9.6.3.3.3

Rewrite the expression.

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - \frac{5}{\frac{{(- 10)}^{\frac{1}{13333}}}{{(- 10000)}^{\frac{1}{13333}}}}$

Step 9.7

Multiply the numerator by the reciprocal of the denominator.

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - (5 \frac{{(- 10000)}^{\frac{1}{13333}}}{{(- 10)}^{\frac{1}{13333}}})$

Step 9.8

Use the power of quotient rule $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - (5 {(\frac{- 10000}{- 10})}^{\frac{1}{13333}})$

Step 9.9

Divide $- 10000$ by $- 10$ .

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - (5 \cdot 1000^{\frac{1}{13333}})$

Step 9.10

Multiply $5$ by $- 1$ .

$- \frac{199990000 {(- 10)}^{\frac{13332}{13333}}}{{(- 10000)}^{\frac{13332}{13333}}} - 5 \cdot 1000^{\frac{1}{13333}}$

Step 10

$x = \frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}$ is a local maximum

Step 11

Find the y-value when $x = \frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}$ .

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Step 11.1

Replace the variable $x$ with $\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}$ in the expression.

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = 20 {(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{\frac{1}{2}} - \frac{1}{2} \cdot {(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{20000}$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Apply the product rule to $\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = 20 (\frac{{({(- 10)}^{\frac{2}{39999}})}^{\frac{1}{2}}}{{({(- 10000)}^{\frac{2}{39999}})}^{\frac{1}{2}}}) - \frac{1}{2} \cdot {(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{20000}$

Step 11.2.1.2

Multiply the exponents in ${({(- 10)}^{\frac{2}{39999}})}^{\frac{1}{2}}$ .

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Step 11.2.1.2.1

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

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = 20 (\frac{{(- 10)}^{\frac{2}{39999} \cdot \frac{1}{2}}}{{({(- 10000)}^{\frac{2}{39999}})}^{\frac{1}{2}}}) - \frac{1}{2} \cdot {(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{20000}$

Step 11.2.1.2.2

Cancel the common factor of $2$ .

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Step 11.2.1.2.2.1

Cancel the common factor.

Step 11.2.1.2.2.2

Rewrite the expression.

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = 20 (\frac{{(- 10)}^{\frac{1}{39999}}}{{({(- 10000)}^{\frac{2}{39999}})}^{\frac{1}{2}}}) - \frac{1}{2} \cdot {(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{20000}$

Step 11.2.1.3

Multiply the exponents in ${({(- 10000)}^{\frac{2}{39999}})}^{\frac{1}{2}}$ .

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Step 11.2.1.3.1

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

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = 20 (\frac{{(- 10)}^{\frac{1}{39999}}}{{(- 10000)}^{\frac{2}{39999} \cdot \frac{1}{2}}}) - \frac{1}{2} \cdot {(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{20000}$

Step 11.2.1.3.2

Cancel the common factor of $2$ .

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Step 11.2.1.3.2.1

Cancel the common factor.

Step 11.2.1.3.2.2

Rewrite the expression.

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = 20 (\frac{{(- 10)}^{\frac{1}{39999}}}{{(- 10000)}^{\frac{1}{39999}}}) - \frac{1}{2} \cdot {(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{20000}$

Step 11.2.1.4

Combine $20$ and $\frac{{(- 10)}^{\frac{1}{39999}}}{{(- 10000)}^{\frac{1}{39999}}}$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}}}{{(- 10000)}^{\frac{1}{39999}}} - \frac{1}{2} \cdot {(\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}})}^{20000}$

Step 11.2.1.5

Apply the product rule to $\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}}}{{(- 10000)}^{\frac{1}{39999}}} - \frac{1}{2} \cdot \frac{{({(- 10)}^{\frac{2}{39999}})}^{20000}}{{({(- 10000)}^{\frac{2}{39999}})}^{20000}}$

Step 11.2.1.6

Multiply the exponents in ${({(- 10)}^{\frac{2}{39999}})}^{20000}$ .

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Step 11.2.1.6.1

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

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}}}{{(- 10000)}^{\frac{1}{39999}}} - \frac{1}{2} \cdot \frac{{(- 10)}^{\frac{2}{39999} \cdot 20000}}{{({(- 10000)}^{\frac{2}{39999}})}^{20000}}$

Step 11.2.1.6.2

Multiply $\frac{2}{39999} \cdot 20000$ .

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Step 11.2.1.6.2.1

Combine $\frac{2}{39999}$ and $20000$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}}}{{(- 10000)}^{\frac{1}{39999}}} - \frac{1}{2} \cdot \frac{{(- 10)}^{\frac{2 \cdot 20000}{39999}}}{{({(- 10000)}^{\frac{2}{39999}})}^{20000}}$

Step 11.2.1.6.2.2

Multiply $2$ by $20000$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}}}{{(- 10000)}^{\frac{1}{39999}}} - \frac{1}{2} \cdot \frac{{(- 10)}^{\frac{40000}{39999}}}{{({(- 10000)}^{\frac{2}{39999}})}^{20000}}$

Step 11.2.1.7

Multiply the exponents in ${({(- 10000)}^{\frac{2}{39999}})}^{20000}$ .

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Step 11.2.1.7.1

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

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}}}{{(- 10000)}^{\frac{1}{39999}}} - \frac{1}{2} \cdot \frac{{(- 10)}^{\frac{40000}{39999}}}{{(- 10000)}^{\frac{2}{39999} \cdot 20000}}$

Step 11.2.1.7.2

Multiply $\frac{2}{39999} \cdot 20000$ .

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Step 11.2.1.7.2.1

Combine $\frac{2}{39999}$ and $20000$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}}}{{(- 10000)}^{\frac{1}{39999}}} - \frac{1}{2} \cdot \frac{{(- 10)}^{\frac{40000}{39999}}}{{(- 10000)}^{\frac{2 \cdot 20000}{39999}}}$

Step 11.2.1.7.2.2

Multiply $2$ by $20000$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}}}{{(- 10000)}^{\frac{1}{39999}}} - \frac{1}{2} \cdot \frac{{(- 10)}^{\frac{40000}{39999}}}{{(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.1.8

Multiply $\frac{{(- 10)}^{\frac{40000}{39999}}}{{(- 10000)}^{\frac{40000}{39999}}}$ by $\frac{1}{2}$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}}}{{(- 10000)}^{\frac{1}{39999}}} - \frac{{(- 10)}^{\frac{40000}{39999}}}{{(- 10000)}^{\frac{40000}{39999}} \cdot 2}$

Step 11.2.1.9

Move $2$ to the left of ${(- 10000)}^{\frac{40000}{39999}}$ .

Step 11.2.2

To write $\frac{20 {(- 10)}^{\frac{1}{39999}}}{{(- 10000)}^{\frac{1}{39999}}}$ as a fraction with a common denominator, multiply by $\frac{2 {(- 10000)}^{\frac{39999}{39999}}}{2 {(- 10000)}^{\frac{39999}{39999}}}$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}}}{{(- 10000)}^{\frac{1}{39999}}} \cdot \frac{2 {(- 10000)}^{\frac{39999}{39999}}}{2 {(- 10000)}^{\frac{39999}{39999}}} - \frac{{(- 10)}^{\frac{40000}{39999}}}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.3

Write each expression with a common denominator of $2 {(- 10000)}^{\frac{40000}{39999}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.2.3.1

Multiply $\frac{20 {(- 10)}^{\frac{1}{39999}}}{{(- 10000)}^{\frac{1}{39999}}}$ by $\frac{2 {(- 10000)}^{\frac{39999}{39999}}}{2 {(- 10000)}^{\frac{39999}{39999}}}$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}} (2 {(- 10000)}^{\frac{39999}{39999}})}{{(- 10000)}^{\frac{1}{39999}} (2 {(- 10000)}^{\frac{39999}{39999}})} - \frac{{(- 10)}^{\frac{40000}{39999}}}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.3.2

Multiply ${(- 10000)}^{\frac{1}{39999}}$ by ${(- 10000)}^{\frac{39999}{39999}}$ by adding the exponents.

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Step 11.2.3.2.1

Move ${(- 10000)}^{\frac{39999}{39999}}$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}} (2 {(- 10000)}^{\frac{39999}{39999}})}{{(- 10000)}^{\frac{39999}{39999}} {(- 10000)}^{\frac{1}{39999}} \cdot 2} - \frac{{(- 10)}^{\frac{40000}{39999}}}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.3.2.2

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

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}} (2 {(- 10000)}^{\frac{39999}{39999}})}{{(- 10000)}^{\frac{39999}{39999} + \frac{1}{39999}} \cdot 2} - \frac{{(- 10)}^{\frac{40000}{39999}}}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.3.2.3

Combine the numerators over the common denominator.

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}} (2 {(- 10000)}^{\frac{39999}{39999}})}{{(- 10000)}^{\frac{39999 + 1}{39999}} \cdot 2} - \frac{{(- 10)}^{\frac{40000}{39999}}}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.3.2.4

Add $39999$ and $1$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}} (2 {(- 10000)}^{\frac{39999}{39999}})}{{(- 10000)}^{\frac{40000}{39999}} \cdot 2} - \frac{{(- 10)}^{\frac{40000}{39999}}}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.3.3

Reorder the factors of ${(- 10000)}^{\frac{40000}{39999}} \cdot 2$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}} (2 {(- 10000)}^{\frac{39999}{39999}})}{2 {(- 10000)}^{\frac{40000}{39999}}} - \frac{{(- 10)}^{\frac{40000}{39999}}}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.4

Reduce the expression by cancelling the common factors.

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Step 11.2.4.1

Combine the numerators over the common denominator.

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}} (2 {(- 10000)}^{\frac{39999}{39999}}) - {(- 10)}^{\frac{40000}{39999}}}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.4.2

Cancel the common factor of $39999$ .

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Step 11.2.4.2.1

Cancel the common factor.

Step 11.2.4.2.2

Rewrite the expression.

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{20 {(- 10)}^{\frac{1}{39999}} (2 (- 10000)) - {(- 10)}^{\frac{40000}{39999}}}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.5

Simplify the numerator.

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Step 11.2.5.1

Multiply $2$ by $20$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{40 {(- 10)}^{\frac{1}{39999}} (- 10000) - {(- 10)}^{\frac{40000}{39999}}}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.5.2

Evaluate the exponent.

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{40 {(- 10)}^{\frac{1}{39999}} \cdot - 10000 - {(- 10)}^{\frac{40000}{39999}}}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.5.3

Multiply $- 10000$ by $40$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{- 400000 {(- 10)}^{\frac{1}{39999}} - {(- 10)}^{\frac{40000}{39999}}}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.6

Simplify with factoring out.

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Step 11.2.6.1

Factor $- 1$ out of $- 400000 {(- 10)}^{\frac{1}{39999}}$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{- (400000 {(- 10)}^{\frac{1}{39999}}) - {(- 10)}^{\frac{40000}{39999}}}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.6.2

Factor $- 1$ out of $- {(- 10)}^{\frac{40000}{39999}}$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{- (400000 {(- 10)}^{\frac{1}{39999}}) - ({(- 10)}^{\frac{40000}{39999}})}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.6.3

Factor $- 1$ out of $- (400000 {(- 10)}^{\frac{1}{39999}}) - ({(- 10)}^{\frac{40000}{39999}})$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{- (400000 {(- 10)}^{\frac{1}{39999}} + {(- 10)}^{\frac{40000}{39999}})}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.6.4

Simplify the expression.

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Step 11.2.6.4.1

Rewrite $- (400000 {(- 10)}^{\frac{1}{39999}} + {(- 10)}^{\frac{40000}{39999}})$ as $- 1 (400000 {(- 10)}^{\frac{1}{39999}} + {(- 10)}^{\frac{40000}{39999}})$ .

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = \frac{- 1 (400000 {(- 10)}^{\frac{1}{39999}} + {(- 10)}^{\frac{40000}{39999}})}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.6.4.2

Move the negative in front of the fraction.

$f (\frac{{(- 10)}^{\frac{2}{39999}}}{{(- 10000)}^{\frac{2}{39999}}}) = - \frac{400000 {(- 10)}^{\frac{1}{39999}} + {(- 10)}^{\frac{40000}{39999}}}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 11.2.7

The final answer is $- \frac{400000 {(- 10)}^{\frac{1}{39999}} + {(- 10)}^{\frac{40000}{39999}}}{2 {(- 10000)}^{\frac{40000}{39999}}}$ .

$y = - \frac{400000 {(- 10)}^{\frac{1}{39999}} + {(- 10)}^{\frac{40000}{39999}}}{2 {(- 10000)}^{\frac{40000}{39999}}}$

Step 12

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.

$- 199990000 {(0)}^{19998} - \frac{5}{{(0)}^{\frac{3}{2}}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify the expression.

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Step 13.1.1

Rewrite $0$ as $0^{2}$ .

$- 199990000 {(0)}^{19998} - \frac{5}{{(0^{2})}^{\frac{3}{2}}}$

Step 13.1.2

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

$- 199990000 {(0)}^{19998} - \frac{5}{0^{2 (\frac{3}{2})}}$

Step 13.2

Cancel the common factor of $2$ .

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Step 13.2.1

Cancel the common factor.

$- 199990000 {(0)}^{19998} - \frac{5}{0^{2 (\frac{3}{2})}}$

Step 13.2.2

Rewrite the expression.

$- 199990000 {(0)}^{19998} - \frac{5}{0^{3}}$

Step 13.3

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

$- 199990000 {(0)}^{19998} - \frac{5}{0}$

Step 13.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 14

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 15