Calculus Examples

$4 x - \frac{4000}{x^{2}}$

Step 1

Write $4 x - \frac{4000}{x^{2}}$ as a function.

$f (x) = 4 x - \frac{4000}{x^{2}}$

Step 2

Find the first derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $4 x - \frac{4000}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- \frac{4000}{x^{2}}]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [- \frac{4000}{x^{2}}]$

Step 2.2

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 2.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{4000}{x^{2}}]$

Step 2.2.2

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

$4 \cdot 1 + \frac{d}{d x} [- \frac{4000}{x^{2}}]$

Step 2.2.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [- \frac{4000}{x^{2}}]$

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{4000}{x^{2}}]$ .

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Step 2.3.1

Since $- 4000$ is constant with respect to $x$ , the derivative of $- \frac{4000}{x^{2}}$ with respect to $x$ is $- 4000 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$4 - 4000 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 2.3.2

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

$4 - 4000 \frac{d}{d x} [{(x^{2})}^{- 1}]$

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

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Step 2.3.3.1

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

$4 - 4000 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}])$

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 $n = - 1$ .

$4 - 4000 (- u^{- 2} \frac{d}{d x} [x^{2}])$

Step 2.3.3.3

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

$4 - 4000 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}])$

Step 2.3.4

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

$4 - 4000 (- {(x^{2})}^{- 2} (2 x))$

Step 2.3.5

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

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Step 2.3.5.1

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

$4 - 4000 (- x^{2 \cdot - 2} (2 x))$

Step 2.3.5.2

Multiply $2$ by $- 2$ .

$4 - 4000 (- x^{- 4} (2 x))$

Step 2.3.6

Multiply $2$ by $- 1$ .

$4 - 4000 (- 2 x^{- 4} x)$

Step 2.3.7

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

$4 - 4000 (- 2 (x^{1} x^{- 4}))$

Step 2.3.8

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

$4 - 4000 (- 2 x^{1 - 4})$

Step 2.3.9

Subtract $4$ from $1$ .

$4 - 4000 (- 2 x^{- 3})$

Step 2.3.10

Multiply $- 2$ by $- 4000$ .

$4 + 8000 x^{- 3}$

Step 2.4

Simplify.

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Step 2.4.1

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

$4 + 8000 \frac{1}{x^{3}}$

Step 2.4.2

Combine $8000$ and $\frac{1}{x^{3}}$ .

$4 + \frac{8000}{x^{3}}$

Step 2.4.3

Reorder terms.

$\frac{8000}{x^{3}} + 4$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $\frac{8000}{x^{3}} + 4$ with respect to $x$ is $\frac{d}{d x} [\frac{8000}{x^{3}}] + \frac{d}{d x} [4]$ .

$f'' (x) = \frac{d}{d x} (\frac{8000}{x^{3}}) + \frac{d}{d x} (4)$

Step 3.2

Evaluate $\frac{d}{d x} [\frac{8000}{x^{3}}]$ .

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Step 3.2.1

Since $8000$ is constant with respect to $x$ , the derivative of $\frac{8000}{x^{3}}$ with respect to $x$ is $8000 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$f'' (x) = 8000 \frac{d}{d x} (\frac{1}{x^{3}}) + \frac{d}{d x} (4)$

Step 3.2.2

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

$f'' (x) = 8000 \frac{d}{d x} ({(x^{3})}^{- 1}) + \frac{d}{d x} (4)$

Step 3.2.3

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 $g (x) = x^{3}$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u$ as $x^{3}$ .

$f'' (x) = 8000 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{3})) + \frac{d}{d x} (4)$

Step 3.2.3.2

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

$f'' (x) = 8000 (- u^{- 2} \frac{d}{d x} x^{3}) + \frac{d}{d x} (4)$

Step 3.2.3.3

Replace all occurrences of $u$ with $x^{3}$ .

$f'' (x) = 8000 (- {(x^{3})}^{- 2} \frac{d}{d x} x^{3}) + \frac{d}{d x} (4)$

Step 3.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = 8000 (- {(x^{3})}^{- 2} (3 x^{2})) + \frac{d}{d x} (4)$

Step 3.2.5

Multiply the exponents in ${(x^{3})}^{- 2}$ .

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Step 3.2.5.1

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

$f'' (x) = 8000 (- x^{3 \cdot - 2} (3 x^{2})) + \frac{d}{d x} (4)$

Step 3.2.5.2

Multiply $3$ by $- 2$ .

$f'' (x) = 8000 (- x^{- 6} (3 x^{2})) + \frac{d}{d x} (4)$

Step 3.2.6

Multiply $3$ by $- 1$ .

$f'' (x) = 8000 (- 3 x^{- 6} x^{2}) + \frac{d}{d x} (4)$

Step 3.2.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 3.2.7.1

Move $x^{2}$ .

$f'' (x) = 8000 (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} (4)$

Step 3.2.7.2

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

$f'' (x) = 8000 (- 3 x^{2 - 6}) + \frac{d}{d x} (4)$

Step 3.2.7.3

Subtract $6$ from $2$ .

$f'' (x) = 8000 (- 3 x^{- 4}) + \frac{d}{d x} (4)$

Step 3.2.8

Multiply $- 3$ by $8000$ .

$f'' (x) = - 24000 x^{- 4} + \frac{d}{d x} (4)$

Step 3.3

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$f'' (x) = - 24000 x^{- 4} + 0$

Step 3.4

Simplify.

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Step 3.4.1

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

$f'' (x) = - 24000 \frac{1}{x^{4}} + 0$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Combine $- 24000$ and $\frac{1}{x^{4}}$ .

$f'' (x) = \frac{- 24000}{x^{4}} + 0$

Step 3.4.2.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{24000}{x^{4}} + 0$

Step 3.4.2.3

Add $- \frac{24000}{x^{4}}$ and $0$ .

$f'' (x) = - \frac{24000}{x^{4}}$

Step 4

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

$\frac{8000}{x^{3}} + 4 = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

By the Sum Rule, the derivative of $4 x - \frac{4000}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- \frac{4000}{x^{2}}]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [- \frac{4000}{x^{2}}]$

Step 5.1.2

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 5.1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{4000}{x^{2}}]$

Step 5.1.2.2

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

$4 \cdot 1 + \frac{d}{d x} [- \frac{4000}{x^{2}}]$

Step 5.1.2.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [- \frac{4000}{x^{2}}]$

Step 5.1.3

Evaluate $\frac{d}{d x} [- \frac{4000}{x^{2}}]$ .

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Step 5.1.3.1

Since $- 4000$ is constant with respect to $x$ , the derivative of $- \frac{4000}{x^{2}}$ with respect to $x$ is $- 4000 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$4 - 4000 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 5.1.3.2

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

$4 - 4000 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 5.1.3.3

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

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Step 5.1.3.3.1

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

$4 - 4000 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}])$

Step 5.1.3.3.2

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

$4 - 4000 (- u^{- 2} \frac{d}{d x} [x^{2}])$

Step 5.1.3.3.3

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

$4 - 4000 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}])$

Step 5.1.3.4

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

$4 - 4000 (- {(x^{2})}^{- 2} (2 x))$

Step 5.1.3.5

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

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Step 5.1.3.5.1

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

$4 - 4000 (- x^{2 \cdot - 2} (2 x))$

Step 5.1.3.5.2

Multiply $2$ by $- 2$ .

$4 - 4000 (- x^{- 4} (2 x))$

Step 5.1.3.6

Multiply $2$ by $- 1$ .

$4 - 4000 (- 2 x^{- 4} x)$

Step 5.1.3.7

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

$4 - 4000 (- 2 (x^{1} x^{- 4}))$

Step 5.1.3.8

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

$4 - 4000 (- 2 x^{1 - 4})$

Step 5.1.3.9

Subtract $4$ from $1$ .

$4 - 4000 (- 2 x^{- 3})$

Step 5.1.3.10

Multiply $- 2$ by $- 4000$ .

$4 + 8000 x^{- 3}$

Step 5.1.4

Simplify.

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Step 5.1.4.1

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

$4 + 8000 \frac{1}{x^{3}}$

Step 5.1.4.2

Combine $8000$ and $\frac{1}{x^{3}}$ .

$4 + \frac{8000}{x^{3}}$

Step 5.1.4.3

Reorder terms.

$f' (x) = \frac{8000}{x^{3}} + 4$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{8000}{x^{3}} + 4$ .

$\frac{8000}{x^{3}} + 4$

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{8000}{x^{3}} + 4 = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$\frac{8000}{x^{3}} + 4 = 0$

Step 6.2

Subtract $4$ from both sides of the equation.

$\frac{8000}{x^{3}} = - 4$

Step 6.3

Find the LCD of the terms in the equation.

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Step 6.3.1

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

$x^{3}, 1$

Step 6.3.2

The LCM of one and any expression is the expression.

$x^{3}$

Step 6.4

Multiply each term in $\frac{8000}{x^{3}} = - 4$ by $x^{3}$ to eliminate the fractions.

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Step 6.4.1

Multiply each term in $\frac{8000}{x^{3}} = - 4$ by $x^{3}$ .

$\frac{8000}{x^{3}} x^{3} = - 4 x^{3}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $x^{3}$ .

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Step 6.4.2.1.1

Cancel the common factor.

$\frac{8000}{x^{3}} x^{3} = - 4 x^{3}$

Step 6.4.2.1.2

Rewrite the expression.

$8000 = - 4 x^{3}$

Step 6.5

Solve the equation.

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Step 6.5.1

Rewrite the equation as $- 4 x^{3} = 8000$ .

$- 4 x^{3} = 8000$

Step 6.5.2

Subtract $8000$ from both sides of the equation.

$- 4 x^{3} - 8000 = 0$

Step 6.5.3

Factor $- 4$ out of $- 4 x^{3} - 8000$ .

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Step 6.5.3.1

Factor $- 4$ out of $- 4 x^{3}$ .

$- 4 x^{3} - 8000 = 0$

Step 6.5.3.2

Factor $- 4$ out of $- 8000$ .

$- 4 x^{3} - 4 \cdot 2000 = 0$

Step 6.5.3.3

Factor $- 4$ out of $- 4 (x^{3}) - 4 (2000)$ .

$- 4 (x^{3} + 2000) = 0$

Step 6.5.4

Divide each term in $- 4 (x^{3} + 2000) = 0$ by $- 4$ and simplify.

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Step 6.5.4.1

Divide each term in $- 4 (x^{3} + 2000) = 0$ by $- 4$ .

$\frac{- 4 (x^{3} + 2000)}{- 4} = \frac{0}{- 4}$

Step 6.5.4.2

Simplify the left side.

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Step 6.5.4.2.1

Cancel the common factor of $- 4$ .

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Step 6.5.4.2.1.1

Cancel the common factor.

$\frac{- 4 (x^{3} + 2000)}{- 4} = \frac{0}{- 4}$

Step 6.5.4.2.1.2

Divide $x^{3} + 2000$ by $1$ .

$x^{3} + 2000 = \frac{0}{- 4}$

Step 6.5.4.3

Simplify the right side.

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Step 6.5.4.3.1

Divide $0$ by $- 4$ .

$x^{3} + 2000 = 0$

Step 6.5.5

Subtract $2000$ from both sides of the equation.

$x^{3} = - 2000$

Step 6.5.6

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

$x = \sqrt[3]{- 2000}$

Step 6.5.7

Simplify $\sqrt[3]{- 2000}$ .

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Step 6.5.7.1

Rewrite $- 2000$ as ${(- 10)}^{3} \cdot 2$ .

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Step 6.5.7.1.1

Factor $- 1000$ out of $- 2000$ .

$x = \sqrt[3]{- 1000 (2)}$

Step 6.5.7.1.2

Rewrite $- 1000$ as ${(- 10)}^{3}$ .

$x = \sqrt[3]{{(- 10)}^{3} \cdot 2}$

Step 6.5.7.2

Pull terms out from under the radical.

$x = - 10 \sqrt[3]{2}$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Set the denominator in $\frac{8000}{x^{3}}$ equal to $0$ to find where the expression is undefined.

$x^{3} = 0$

Step 7.2

Solve for $x$ .

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Step 7.2.1

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

$x = \sqrt[3]{0}$

Step 7.2.2

Simplify $\sqrt[3]{0}$ .

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Step 7.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 7.2.2.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 8

Critical points to evaluate.

$x = - 10 \sqrt[3]{2}$

Step 9

Evaluate the second derivative at $x = - 10 \sqrt[3]{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{24000}{{(- 10 \sqrt[3]{2})}^{4}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the denominator.

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Step 10.1.1

Apply the product rule to $- 10 \sqrt[3]{2}$ .

$- \frac{24000}{{(- 10)}^{4} {\sqrt[3]{2}}^{4}}$

Step 10.1.2

Raise $- 10$ to the power of $4$ .

$- \frac{24000}{10000 {\sqrt[3]{2}}^{4}}$

Step 10.1.3

Rewrite ${\sqrt[3]{2}}^{4}$ as $\sqrt[3]{2^{4}}$ .

$- \frac{24000}{10000 \sqrt[3]{2^{4}}}$

Step 10.1.4

Raise $2$ to the power of $4$ .

$- \frac{24000}{10000 \sqrt[3]{16}}$

Step 10.1.5

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 10.1.5.1

Factor $8$ out of $16$ .

$- \frac{24000}{10000 \sqrt[3]{8 (2)}}$

Step 10.1.5.2

Rewrite $8$ as $2^{3}$ .

$- \frac{24000}{10000 \sqrt[3]{2^{3} \cdot 2}}$

Step 10.1.6

Pull terms out from under the radical.

$- \frac{24000}{10000 \cdot 2 \sqrt[3]{2}}$

Step 10.1.7

Multiply $10000$ by $2$ .

$- \frac{24000}{20000 \sqrt[3]{2}}$

Step 10.2

Cancel the common factor of $24000$ and $20000$ .

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Step 10.2.1

Factor $4000$ out of $24000$ .

$- \frac{4000 (6)}{20000 \sqrt[3]{2}}$

Step 10.2.2

Cancel the common factors.

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Step 10.2.2.1

Factor $4000$ out of $20000 \sqrt[3]{2}$ .

$- \frac{4000 (6)}{4000 (5 \sqrt[3]{2})}$

Step 10.2.2.2

Cancel the common factor.

$- \frac{4000 \cdot 6}{4000 (5 \sqrt[3]{2})}$

Step 10.2.2.3

Rewrite the expression.

$- \frac{6}{5 \sqrt[3]{2}}$

Step 10.3

Multiply $\frac{6}{5 \sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$- (\frac{6}{5 \sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}})$

Step 10.4

Simplify terms.

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Step 10.4.1

Combine and simplify the denominator.

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Step 10.4.1.1

Multiply $\frac{6}{5 \sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$- \frac{6 {\sqrt[3]{2}}^{2}}{5 \sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Step 10.4.1.2

Move $\sqrt[3]{2}$ .

$- \frac{6 {\sqrt[3]{2}}^{2}}{5 (\sqrt[3]{2} {\sqrt[3]{2}}^{2})}$

Step 10.4.1.3

Raise $\sqrt[3]{2}$ to the power of $1$ .

$- \frac{6 {\sqrt[3]{2}}^{2}}{5 ({\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2})}$

Step 10.4.1.4

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

$- \frac{6 {\sqrt[3]{2}}^{2}}{5 {\sqrt[3]{2}}^{1 + 2}}$

Step 10.4.1.5

Add $1$ and $2$ .

$- \frac{6 {\sqrt[3]{2}}^{2}}{5 {\sqrt[3]{2}}^{3}}$

Step 10.4.1.6

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

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Step 10.4.1.6.1

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

$- \frac{6 {\sqrt[3]{2}}^{2}}{5 {(2^{\frac{1}{3}})}^{3}}$

Step 10.4.1.6.2

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

$- \frac{6 {\sqrt[3]{2}}^{2}}{5 \cdot 2^{\frac{1}{3} \cdot 3}}$

Step 10.4.1.6.3

Combine $\frac{1}{3}$ and $3$ .

$- \frac{6 {\sqrt[3]{2}}^{2}}{5 \cdot 2^{\frac{3}{3}}}$

Step 10.4.1.6.4

Cancel the common factor of $3$ .

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Step 10.4.1.6.4.1

Cancel the common factor.

$- \frac{6 {\sqrt[3]{2}}^{2}}{5 \cdot 2^{\frac{3}{3}}}$

Step 10.4.1.6.4.2

Rewrite the expression.

$- \frac{6 {\sqrt[3]{2}}^{2}}{5 \cdot 2^{1}}$

Step 10.4.1.6.5

Evaluate the exponent.

$- \frac{6 {\sqrt[3]{2}}^{2}}{5 \cdot 2}$

Step 10.4.2

Cancel the common factor of $6$ and $2$ .

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Step 10.4.2.1

Factor $2$ out of $6 {\sqrt[3]{2}}^{2}$ .

$- \frac{2 (3 {\sqrt[3]{2}}^{2})}{5 \cdot 2}$

Step 10.4.2.2

Cancel the common factors.

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Step 10.4.2.2.1

Factor $2$ out of $5 \cdot 2$ .

$- \frac{2 (3 {\sqrt[3]{2}}^{2})}{2 \cdot 5}$

Step 10.4.2.2.2

Cancel the common factor.

$- \frac{2 (3 {\sqrt[3]{2}}^{2})}{2 \cdot 5}$

Step 10.4.2.2.3

Rewrite the expression.

$- \frac{3 {\sqrt[3]{2}}^{2}}{5}$

Step 10.5

Simplify the numerator.

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Step 10.5.1

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

$- \frac{3 \sqrt[3]{2^{2}}}{5}$

Step 10.5.2

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

$- \frac{3 \sqrt[3]{4}}{5}$

Step 11

$x = - 10 \sqrt[3]{2}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - 10 \sqrt[3]{2}$ is a local maximum

Step 12

Find the y-value when $x = - 10 \sqrt[3]{2}$ .

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Step 12.1

Replace the variable $x$ with $- 10 \sqrt[3]{2}$ in the expression.

$f (- 10 \sqrt[3]{2}) = 4 (- 10 \sqrt[3]{2}) - \frac{4000}{{(- 10 \sqrt[3]{2})}^{2}}$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Multiply $- 10$ by $4$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{4000}{{(- 10 \sqrt[3]{2})}^{2}}$

Step 12.2.1.2

Simplify the denominator.

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Step 12.2.1.2.1

Apply the product rule to $- 10 \sqrt[3]{2}$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{4000}{{(- 10)}^{2} {\sqrt[3]{2}}^{2}}$

Step 12.2.1.2.2

Raise $- 10$ to the power of $2$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{4000}{100 {\sqrt[3]{2}}^{2}}$

Step 12.2.1.2.3

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

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{4000}{100 \sqrt[3]{2^{2}}}$

Step 12.2.1.2.4

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

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{4000}{100 \sqrt[3]{4}}$

Step 12.2.1.3

Cancel the common factor of $4000$ and $100$ .

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Step 12.2.1.3.1

Factor $100$ out of $4000$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{100 \cdot 40}{100 \sqrt[3]{4}}$

Step 12.2.1.3.2

Cancel the common factors.

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Step 12.2.1.3.2.1

Factor $100$ out of $100 \sqrt[3]{4}$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{100 \cdot 40}{100 (\sqrt[3]{4})}$

Step 12.2.1.3.2.2

Cancel the common factor.

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{100 \cdot 40}{100 \sqrt[3]{4}}$

Step 12.2.1.3.2.3

Rewrite the expression.

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{40}{\sqrt[3]{4}}$

Step 12.2.1.4

Multiply $\frac{40}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - (\frac{40}{\sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}})$

Step 12.2.1.5

Combine and simplify the denominator.

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Step 12.2.1.5.1

Multiply $\frac{40}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{40 {\sqrt[3]{4}}^{2}}{\sqrt[3]{4} {\sqrt[3]{4}}^{2}}$

Step 12.2.1.5.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{40 {\sqrt[3]{4}}^{2}}{\sqrt[3]{4} {\sqrt[3]{4}}^{2}}$

Step 12.2.1.5.3

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

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{40 {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}}$

Step 12.2.1.5.4

Add $1$ and $2$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{40 {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{3}}$

Step 12.2.1.5.5

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

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Step 12.2.1.5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{40 {\sqrt[3]{4}}^{2}}{{(4^{\frac{1}{3}})}^{3}}$

Step 12.2.1.5.5.2

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

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{40 {\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}}$

Step 12.2.1.5.5.3

Combine $\frac{1}{3}$ and $3$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{40 {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 12.2.1.5.5.4

Cancel the common factor of $3$ .

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Step 12.2.1.5.5.4.1

Cancel the common factor.

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{40 {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 12.2.1.5.5.4.2

Rewrite the expression.

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{40 {\sqrt[3]{4}}^{2}}{4}$

Step 12.2.1.5.5.5

Evaluate the exponent.

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{40 {\sqrt[3]{4}}^{2}}{4}$

Step 12.2.1.6

Cancel the common factor of $40$ and $4$ .

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Step 12.2.1.6.1

Factor $4$ out of $40 {\sqrt[3]{4}}^{2}$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{4 (10 {\sqrt[3]{4}}^{2})}{4}$

Step 12.2.1.6.2

Cancel the common factors.

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Step 12.2.1.6.2.1

Factor $4$ out of $4$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{4 (10 {\sqrt[3]{4}}^{2})}{4 (1)}$

Step 12.2.1.6.2.2

Cancel the common factor.

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{4 (10 {\sqrt[3]{4}}^{2})}{4 \cdot 1}$

Step 12.2.1.6.2.3

Rewrite the expression.

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - \frac{10 {\sqrt[3]{4}}^{2}}{1}$

Step 12.2.1.6.2.4

Divide $10 {\sqrt[3]{4}}^{2}$ by $1$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - (10 {\sqrt[3]{4}}^{2})$

Step 12.2.1.7

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - (10 \sqrt[3]{4^{2}})$

Step 12.2.1.8

Raise $4$ to the power of $2$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - (10 \sqrt[3]{16})$

Step 12.2.1.9

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 12.2.1.9.1

Factor $8$ out of $16$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - (10 \sqrt[3]{8 (2)})$

Step 12.2.1.9.2

Rewrite $8$ as $2^{3}$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - (10 \sqrt[3]{2^{3} \cdot 2})$

Step 12.2.1.10

Pull terms out from under the radical.

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - (10 (2 \sqrt[3]{2}))$

Step 12.2.1.11

Multiply $2$ by $10$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - (20 \sqrt[3]{2})$

Step 12.2.1.12

Multiply $20$ by $- 1$ .

$f (- 10 \sqrt[3]{2}) = - 40 \sqrt[3]{2} - 20 \sqrt[3]{2}$

Step 12.2.2

Subtract $20 \sqrt[3]{2}$ from $- 40 \sqrt[3]{2}$ .

$f (- 10 \sqrt[3]{2}) = - 60 \sqrt[3]{2}$

Step 12.2.3

The final answer is $- 60 \sqrt[3]{2}$ .

$y = - 60 \sqrt[3]{2}$

Step 13

These are the local extrema for $f (x) = 4 x - \frac{4000}{x^{2}}$ .

$(- 10 \sqrt[3]{2}, - 60 \sqrt[3]{2})$ is a local maxima

Step 14