Algebra Examples

$\frac{2 x^{3} + 28000}{x}$

Step 1

Write $\frac{2 x^{3} + 28000}{x}$ as a function.

$f (x) = \frac{2 x^{3} + 28000}{x}$

Step 2

Find the first derivative of the function.

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Step 2.1

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 $f (x) = 2 x^{3} + 28000$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [2 x^{3} + 28000] - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $2 x^{3} + 28000$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [28000]$ .

$\frac{x (\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [28000]) - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$\frac{x (2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [28000]) - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2.3

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

$\frac{x (2 (3 x^{2}) + \frac{d}{d x} [28000]) - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2.4

Multiply $3$ by $2$ .

$\frac{x (6 x^{2} + \frac{d}{d x} [28000]) - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2.5

Since $28000$ is constant with respect to $x$ , the derivative of $28000$ with respect to $x$ is $0$ .

$\frac{x (6 x^{2} + 0) - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2.6

Add $6 x^{2}$ and $0$ .

$\frac{x (6 x^{2}) - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 2.3

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

$\frac{6 (x^{1} x^{2}) - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 2.4

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

$\frac{6 x^{1 + 2} - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 2.5

Add $1$ and $2$ .

$\frac{6 x^{3} - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 2.6

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

$\frac{6 x^{3} - (2 x^{3} + 28000) \cdot 1}{x^{2}}$

Step 2.7

Multiply $- 1$ by $1$ .

$\frac{6 x^{3} - (2 x^{3} + 28000)}{x^{2}}$

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

$\frac{6 x^{3} - (2 x^{3}) - 1 \cdot 28000}{x^{2}}$

Step 2.8.2

Simplify the numerator.

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Step 2.8.2.1

Simplify each term.

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Step 2.8.2.1.1

Multiply $2$ by $- 1$ .

$\frac{6 x^{3} - 2 x^{3} - 1 \cdot 28000}{x^{2}}$

Step 2.8.2.1.2

Multiply $- 1$ by $28000$ .

$\frac{6 x^{3} - 2 x^{3} - 28000}{x^{2}}$

Step 2.8.2.2

Subtract $2 x^{3}$ from $6 x^{3}$ .

$\frac{4 x^{3} - 28000}{x^{2}}$

Step 2.8.3

Factor $4$ out of $4 x^{3} - 28000$ .

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Step 2.8.3.1

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

$\frac{4 (x^{3}) - 28000}{x^{2}}$

Step 2.8.3.2

Factor $4$ out of $- 28000$ .

$\frac{4 x^{3} + 4 \cdot - 7000}{x^{2}}$

Step 2.8.3.3

Factor $4$ out of $4 x^{3} + 4 \cdot - 7000$ .

$\frac{4 (x^{3} - 7000)}{x^{2}}$

Step 3

Find the second derivative of the function.

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Step 3.1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4 (x^{3} - 7000)}{x^{2}}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{x^{3} - 7000}{x^{2}}]$ .

$f'' (x) = 4 \frac{d}{d x} (\frac{x^{3} - 7000}{x^{2}})$

Step 3.2

$f'' (x) = 4 (\frac{x^{2} \frac{d}{d x} (x^{3} - 7000) - (x^{3} - 7000) \frac{d}{d x} x^{2}}{{(x^{2})}^{2}})$

Step 3.3

Differentiate.

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Step 3.3.1

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

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Step 3.3.1.1

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

$f'' (x) = 4 (\frac{x^{2} \frac{d}{d x} (x^{3} - 7000) - (x^{3} - 7000) \frac{d}{d x} x^{2}}{x^{2 \cdot 2}})$

Step 3.3.1.2

Multiply $2$ by $2$ .

$f'' (x) = 4 (\frac{x^{2} \frac{d}{d x} (x^{3} - 7000) - (x^{3} - 7000) \frac{d}{d x} x^{2}}{x^{4}})$

Step 3.3.2

By the Sum Rule, the derivative of $x^{3} - 7000$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7000]$ .

$f'' (x) = 4 (\frac{x^{2} (\frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 7000)) - (x^{3} - 7000) \frac{d}{d x} x^{2}}{x^{4}})$

Step 3.3.3

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

$f'' (x) = 4 (\frac{x^{2} (3 x^{2} + \frac{d}{d x} (- 7000)) - (x^{3} - 7000) \frac{d}{d x} x^{2}}{x^{4}})$

Step 3.3.4

Since $- 7000$ is constant with respect to $x$ , the derivative of $- 7000$ with respect to $x$ is $0$ .

$f'' (x) = 4 (\frac{x^{2} (3 x^{2} + 0) - (x^{3} - 7000) \frac{d}{d x} x^{2}}{x^{4}})$

Step 3.3.5

Add $3 x^{2}$ and $0$ .

$f'' (x) = 4 (\frac{x^{2} (3 x^{2}) - (x^{3} - 7000) \frac{d}{d x} x^{2}}{x^{4}})$

Step 3.4

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

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Step 3.4.1

Move $x^{2}$ .

$f'' (x) = 4 (\frac{x^{2} x^{2} \cdot 3 - (x^{3} - 7000) \frac{d}{d x} x^{2}}{x^{4}})$

Step 3.4.2

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

$f'' (x) = 4 (\frac{x^{2 + 2} \cdot 3 - (x^{3} - 7000) \frac{d}{d x} x^{2}}{x^{4}})$

Step 3.4.3

Add $2$ and $2$ .

$f'' (x) = 4 (\frac{x^{4} \cdot 3 - (x^{3} - 7000) \frac{d}{d x} x^{2}}{x^{4}})$

Step 3.5

Move $3$ to the left of $x^{4}$ .

$f'' (x) = 4 (\frac{3 \cdot x^{4} - (x^{3} - 7000) \frac{d}{d x} x^{2}}{x^{4}})$

Step 3.6

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

$f'' (x) = 4 (\frac{3 x^{4} - (x^{3} - 7000) (2 x)}{x^{4}})$

Step 3.7

Simplify with factoring out.

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Step 3.7.1

Multiply $2$ by $- 1$ .

$f'' (x) = 4 (\frac{3 x^{4} - 2 (x^{3} - 7000) x}{x^{4}})$

Step 3.7.2

Factor $x$ out of $3 x^{4} - 2 (x^{3} - 7000) x$ .

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Step 3.7.2.1

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

$f'' (x) = 4 (\frac{x (3 x^{3}) - 2 (x^{3} - 7000) x}{x^{4}})$

Step 3.7.2.2

Factor $x$ out of $- 2 (x^{3} - 7000) x$ .

$f'' (x) = 4 (\frac{x (3 x^{3}) + x (- 2 (x^{3} - 7000))}{x^{4}})$

Step 3.7.2.3

Factor $x$ out of $x (3 x^{3}) + x (- 2 (x^{3} - 7000))$ .

$f'' (x) = 4 (\frac{x (3 x^{3} - 2 (x^{3} - 7000))}{x^{4}})$

Step 3.8

Cancel the common factors.

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Step 3.8.1

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

$f'' (x) = 4 (\frac{x (3 x^{3} - 2 (x^{3} - 7000))}{x \cdot x^{3}})$

Step 3.8.2

Cancel the common factor.

$f'' (x) = 4 (\frac{x (3 x^{3} - 2 (x^{3} - 7000))}{x \cdot x^{3}})$

Step 3.8.3

Rewrite the expression.

$f'' (x) = 4 (\frac{3 x^{3} - 2 (x^{3} - 7000)}{x^{3}})$

Step 3.9

Combine $4$ and $\frac{3 x^{3} - 2 (x^{3} - 7000)}{x^{3}}$ .

$f'' (x) = \frac{4 (3 x^{3} - 2 (x^{3} - 7000))}{x^{3}}$

Step 3.10

Simplify.

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Step 3.10.1

Apply the distributive property.

$f'' (x) = \frac{4 (3 x^{3} - 2 x^{3} - 2 \cdot - 7000)}{x^{3}}$

Step 3.10.2

Apply the distributive property.

$f'' (x) = \frac{4 (3 x^{3}) + 4 (- 2 x^{3}) + 4 (- 2 \cdot - 7000)}{x^{3}}$

Step 3.10.3

Simplify the numerator.

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Step 3.10.3.1

Simplify each term.

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Step 3.10.3.1.1

Multiply $3$ by $4$ .

$f'' (x) = \frac{12 x^{3} + 4 (- 2 x^{3}) + 4 (- 2 \cdot - 7000)}{x^{3}}$

Step 3.10.3.1.2

Multiply $- 2$ by $4$ .

$f'' (x) = \frac{12 x^{3} - 8 x^{3} + 4 (- 2 \cdot - 7000)}{x^{3}}$

Step 3.10.3.1.3

Multiply $4 (- 2 \cdot - 7000)$ .

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Step 3.10.3.1.3.1

Multiply $- 2$ by $- 7000$ .

$f'' (x) = \frac{12 x^{3} - 8 x^{3} + 4 \cdot 14000}{x^{3}}$

Step 3.10.3.1.3.2

Multiply $4$ by $14000$ .

$f'' (x) = \frac{12 x^{3} - 8 x^{3} + 56000}{x^{3}}$

Step 3.10.3.2

Subtract $8 x^{3}$ from $12 x^{3}$ .

$f'' (x) = \frac{4 x^{3} + 56000}{x^{3}}$

Step 3.10.4

Factor $4$ out of $4 x^{3} + 56000$ .

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Step 3.10.4.1

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

$f'' (x) = \frac{4 (x^{3}) + 56000}{x^{3}}$

Step 3.10.4.2

Factor $4$ out of $56000$ .

$f'' (x) = \frac{4 x^{3} + 4 \cdot 14000}{x^{3}}$

Step 3.10.4.3

Factor $4$ out of $4 x^{3} + 4 \cdot 14000$ .

$f'' (x) = \frac{4 (x^{3} + 14000)}{x^{3}}$

Step 4

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

$\frac{4 (x^{3} - 7000)}{x^{2}} = 0$

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

$\frac{x \frac{d}{d x} [2 x^{3} + 28000] - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.2

Differentiate.

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Step 5.1.2.1

By the Sum Rule, the derivative of $2 x^{3} + 28000$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [28000]$ .

$\frac{x (\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [28000]) - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.2.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$\frac{x (2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [28000]) - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.2.3

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

$\frac{x (2 (3 x^{2}) + \frac{d}{d x} [28000]) - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.2.4

Multiply $3$ by $2$ .

$\frac{x (6 x^{2} + \frac{d}{d x} [28000]) - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.2.5

Since $28000$ is constant with respect to $x$ , the derivative of $28000$ with respect to $x$ is $0$ .

$\frac{x (6 x^{2} + 0) - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.2.6

Add $6 x^{2}$ and $0$ .

$\frac{x (6 x^{2}) - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.3

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

$\frac{6 (x^{1} x^{2}) - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.4

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

$\frac{6 x^{1 + 2} - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.5

Add $1$ and $2$ .

$\frac{6 x^{3} - (2 x^{3} + 28000) \frac{d}{d x} [x]}{x^{2}}$

Step 5.1.6

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

$\frac{6 x^{3} - (2 x^{3} + 28000) \cdot 1}{x^{2}}$

Step 5.1.7

Multiply $- 1$ by $1$ .

$\frac{6 x^{3} - (2 x^{3} + 28000)}{x^{2}}$

Step 5.1.8

Simplify.

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Step 5.1.8.1

Apply the distributive property.

$\frac{6 x^{3} - (2 x^{3}) - 1 \cdot 28000}{x^{2}}$

Step 5.1.8.2

Simplify the numerator.

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Step 5.1.8.2.1

Simplify each term.

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Step 5.1.8.2.1.1

Multiply $2$ by $- 1$ .

$\frac{6 x^{3} - 2 x^{3} - 1 \cdot 28000}{x^{2}}$

Step 5.1.8.2.1.2

Multiply $- 1$ by $28000$ .

$\frac{6 x^{3} - 2 x^{3} - 28000}{x^{2}}$

Step 5.1.8.2.2

Subtract $2 x^{3}$ from $6 x^{3}$ .

$\frac{4 x^{3} - 28000}{x^{2}}$

Step 5.1.8.3

Factor $4$ out of $4 x^{3} - 28000$ .

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Step 5.1.8.3.1

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

$\frac{4 (x^{3}) - 28000}{x^{2}}$

Step 5.1.8.3.2

Factor $4$ out of $- 28000$ .

$\frac{4 x^{3} + 4 \cdot - 7000}{x^{2}}$

Step 5.1.8.3.3

Factor $4$ out of $4 x^{3} + 4 \cdot - 7000$ .

$f' (x) = \frac{4 (x^{3} - 7000)}{x^{2}}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{4 (x^{3} - 7000)}{x^{2}}$ .

$\frac{4 (x^{3} - 7000)}{x^{2}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{4 (x^{3} - 7000)}{x^{2}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$\frac{4 (x^{3} - 7000)}{x^{2}} = 0$

Step 6.2

Set the numerator equal to zero.

$4 (x^{3} - 7000) = 0$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

Divide each term in $4 (x^{3} - 7000) = 0$ by $4$ and simplify.

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Step 6.3.1.1

Divide each term in $4 (x^{3} - 7000) = 0$ by $4$ .

$\frac{4 (x^{3} - 7000)}{4} = \frac{0}{4}$

Step 6.3.1.2

Simplify the left side.

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Step 6.3.1.2.1

Cancel the common factor of $4$ .

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Step 6.3.1.2.1.1

Cancel the common factor.

$\frac{4 (x^{3} - 7000)}{4} = \frac{0}{4}$

Step 6.3.1.2.1.2

Divide $x^{3} - 7000$ by $1$ .

$x^{3} - 7000 = \frac{0}{4}$

Step 6.3.1.3

Simplify the right side.

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Step 6.3.1.3.1

Divide $0$ by $4$ .

$x^{3} - 7000 = 0$

Step 6.3.2

Add $7000$ to both sides of the equation.

$x^{3} = 7000$

Step 6.3.3

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

$x = \sqrt[3]{7000}$

Step 6.3.4

Simplify $\sqrt[3]{7000}$ .

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Step 6.3.4.1

Rewrite $7000$ as $10^{3} \cdot 7$ .

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Step 6.3.4.1.1

Factor $1000$ out of $7000$ .

$x = \sqrt[3]{1000 (7)}$

Step 6.3.4.1.2

Rewrite $1000$ as $10^{3}$ .

$x = \sqrt[3]{10^{3} \cdot 7}$

Step 6.3.4.2

Pull terms out from under the radical.

$x = 10 \sqrt[3]{7}$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Set the denominator in $\frac{4 (x^{3} - 7000)}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

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.

$x = \pm \sqrt{0}$

Step 7.2.2

Simplify $\pm \sqrt{0}$ .

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Step 7.2.2.1

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

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

Step 7.2.2.2

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

$x = \pm 0$

Step 7.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 8

Critical points to evaluate.

$x = 10 \sqrt[3]{7}$

Step 9

Evaluate the second derivative at $x = 10 \sqrt[3]{7}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{4 ({(10 \sqrt[3]{7})}^{3} + 14000)}{{(10 \sqrt[3]{7})}^{3}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Apply the product rule to $10 \sqrt[3]{7}$ .

$\frac{4 (10^{3} {\sqrt[3]{7}}^{3} + 14000)}{{(10 \sqrt[3]{7})}^{3}}$

Step 10.1.2

Raise $10$ to the power of $3$ .

$\frac{4 (1000 {\sqrt[3]{7}}^{3} + 14000)}{{(10 \sqrt[3]{7})}^{3}}$

Step 10.1.3

Rewrite ${\sqrt[3]{7}}^{3}$ as $7$ .

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Step 10.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$\frac{4 (1000 {(7^{\frac{1}{3}})}^{3} + 14000)}{{(10 \sqrt[3]{7})}^{3}}$

Step 10.1.3.2

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

$\frac{4 (1000 \cdot 7^{\frac{1}{3} \cdot 3} + 14000)}{{(10 \sqrt[3]{7})}^{3}}$

Step 10.1.3.3

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

$\frac{4 (1000 \cdot 7^{\frac{3}{3}} + 14000)}{{(10 \sqrt[3]{7})}^{3}}$

Step 10.1.3.4

Cancel the common factor of $3$ .

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Step 10.1.3.4.1

Cancel the common factor.

$\frac{4 (1000 \cdot 7^{\frac{3}{3}} + 14000)}{{(10 \sqrt[3]{7})}^{3}}$

Step 10.1.3.4.2

Rewrite the expression.

$\frac{4 (1000 \cdot 7^{1} + 14000)}{{(10 \sqrt[3]{7})}^{3}}$

Step 10.1.3.5

Evaluate the exponent.

$\frac{4 (1000 \cdot 7 + 14000)}{{(10 \sqrt[3]{7})}^{3}}$

Step 10.1.4

Multiply $1000$ by $7$ .

$\frac{4 (7000 + 14000)}{{(10 \sqrt[3]{7})}^{3}}$

Step 10.1.5

Add $7000$ and $14000$ .

$\frac{4 \cdot 21000}{{(10 \sqrt[3]{7})}^{3}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Apply the product rule to $10 \sqrt[3]{7}$ .

$\frac{4 \cdot 21000}{10^{3} {\sqrt[3]{7}}^{3}}$

Step 10.2.2

Raise $10$ to the power of $3$ .

$\frac{4 \cdot 21000}{1000 {\sqrt[3]{7}}^{3}}$

Step 10.2.3

Rewrite ${\sqrt[3]{7}}^{3}$ as $7$ .

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Step 10.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$\frac{4 \cdot 21000}{1000 {(7^{\frac{1}{3}})}^{3}}$

Step 10.2.3.2

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

$\frac{4 \cdot 21000}{1000 \cdot 7^{\frac{1}{3} \cdot 3}}$

Step 10.2.3.3

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

$\frac{4 \cdot 21000}{1000 \cdot 7^{\frac{3}{3}}}$

Step 10.2.3.4

Cancel the common factor of $3$ .

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Step 10.2.3.4.1

Cancel the common factor.

$\frac{4 \cdot 21000}{1000 \cdot 7^{\frac{3}{3}}}$

Step 10.2.3.4.2

Rewrite the expression.

$\frac{4 \cdot 21000}{1000 \cdot 7^{1}}$

Step 10.2.3.5

Evaluate the exponent.

$\frac{4 \cdot 21000}{1000 \cdot 7}$

Step 10.3

Simplify the expression.

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Step 10.3.1

Multiply $4$ by $21000$ .

$\frac{84000}{1000 \cdot 7}$

Step 10.3.2

Multiply $1000$ by $7$ .

$\frac{84000}{7000}$

Step 10.3.3

Divide $84000$ by $7000$ .

$12$

Step 11

$x = 10 \sqrt[3]{7}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 10 \sqrt[3]{7}$ is a local minimum

Step 12

Find the y-value when $x = 10 \sqrt[3]{7}$ .

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Step 12.1

Replace the variable $x$ with $10 \sqrt[3]{7}$ in the expression.

$f (10 \sqrt[3]{7}) = \frac{2 {(10 \sqrt[3]{7})}^{3} + 28000}{10 \sqrt[3]{7}}$

Step 12.2

Simplify the result.

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Step 12.2.1

Cancel the common factor of $2 {(10 \sqrt[3]{7})}^{3} + 28000$ and $10$ .

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Step 12.2.1.1

Factor $2$ out of $2 {(10 \sqrt[3]{7})}^{3}$ .

$f (10 \sqrt[3]{7}) = \frac{2 ({(10 \sqrt[3]{7})}^{3}) + 28000}{10 \sqrt[3]{7}}$

Step 12.2.1.2

Factor $2$ out of $28000$ .

$f (10 \sqrt[3]{7}) = \frac{2 {(10 \sqrt[3]{7})}^{3} + 2 \cdot 14000}{10 \sqrt[3]{7}}$

Step 12.2.1.3

Factor $2$ out of $2 {(10 \sqrt[3]{7})}^{3} + 2 \cdot 14000$ .

$f (10 \sqrt[3]{7}) = \frac{2 ({(10 \sqrt[3]{7})}^{3} + 14000)}{10 \sqrt[3]{7}}$

Step 12.2.1.4

Cancel the common factors.

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Step 12.2.1.4.1

Factor $2$ out of $10 \sqrt[3]{7}$ .

$f (10 \sqrt[3]{7}) = \frac{2 ({(10 \sqrt[3]{7})}^{3} + 14000)}{2 (5 \sqrt[3]{7})}$

Step 12.2.1.4.2

Cancel the common factor.

$f (10 \sqrt[3]{7}) = \frac{2 ({(10 \sqrt[3]{7})}^{3} + 14000)}{2 (5 \sqrt[3]{7})}$

Step 12.2.1.4.3

Rewrite the expression.

$f (10 \sqrt[3]{7}) = \frac{{(10 \sqrt[3]{7})}^{3} + 14000}{5 \sqrt[3]{7}}$

Step 12.2.2

Simplify the numerator.

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Step 12.2.2.1

Apply the product rule to $10 \sqrt[3]{7}$ .

$f (10 \sqrt[3]{7}) = \frac{10^{3} {\sqrt[3]{7}}^{3} + 14000}{5 \sqrt[3]{7}}$

Step 12.2.2.2

Raise $10$ to the power of $3$ .

$f (10 \sqrt[3]{7}) = \frac{1000 {\sqrt[3]{7}}^{3} + 14000}{5 \sqrt[3]{7}}$

Step 12.2.2.3

Rewrite ${\sqrt[3]{7}}^{3}$ as $7$ .

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Step 12.2.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$f (10 \sqrt[3]{7}) = \frac{1000 {(7^{\frac{1}{3}})}^{3} + 14000}{5 \sqrt[3]{7}}$

Step 12.2.2.3.2

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

$f (10 \sqrt[3]{7}) = \frac{1000 \cdot 7^{\frac{1}{3} \cdot 3} + 14000}{5 \sqrt[3]{7}}$

Step 12.2.2.3.3

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

$f (10 \sqrt[3]{7}) = \frac{1000 \cdot 7^{\frac{3}{3}} + 14000}{5 \sqrt[3]{7}}$

Step 12.2.2.3.4

Cancel the common factor of $3$ .

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Step 12.2.2.3.4.1

Cancel the common factor.

$f (10 \sqrt[3]{7}) = \frac{1000 \cdot 7^{\frac{3}{3}} + 14000}{5 \sqrt[3]{7}}$

Step 12.2.2.3.4.2

Rewrite the expression.

$f (10 \sqrt[3]{7}) = \frac{1000 \cdot 7 + 14000}{5 \sqrt[3]{7}}$

Step 12.2.2.3.5

Evaluate the exponent.

$f (10 \sqrt[3]{7}) = \frac{1000 \cdot 7 + 14000}{5 \sqrt[3]{7}}$

Step 12.2.2.4

Multiply $1000$ by $7$ .

$f (10 \sqrt[3]{7}) = \frac{7000 + 14000}{5 \sqrt[3]{7}}$

Step 12.2.2.5

Add $7000$ and $14000$ .

$f (10 \sqrt[3]{7}) = \frac{21000}{5 \sqrt[3]{7}}$

Step 12.2.3

Cancel the common factor of $21000$ and $5$ .

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Step 12.2.3.1

Factor $5$ out of $21000$ .

$f (10 \sqrt[3]{7}) = \frac{5 \cdot 4200}{5 \sqrt[3]{7}}$

Step 12.2.3.2

Cancel the common factors.

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Step 12.2.3.2.1

Factor $5$ out of $5 \sqrt[3]{7}$ .

$f (10 \sqrt[3]{7}) = \frac{5 \cdot 4200}{5 (\sqrt[3]{7})}$

Step 12.2.3.2.2

Cancel the common factor.

$f (10 \sqrt[3]{7}) = \frac{5 \cdot 4200}{5 \sqrt[3]{7}}$

Step 12.2.3.2.3

Rewrite the expression.

$f (10 \sqrt[3]{7}) = \frac{4200}{\sqrt[3]{7}}$

Step 12.2.4

Multiply $\frac{4200}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$f (10 \sqrt[3]{7}) = \frac{4200}{\sqrt[3]{7}} \cdot \frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$

Step 12.2.5

Combine and simplify the denominator.

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Step 12.2.5.1

Multiply $\frac{4200}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$f (10 \sqrt[3]{7}) = \frac{4200 {\sqrt[3]{7}}^{2}}{\sqrt[3]{7} {\sqrt[3]{7}}^{2}}$

Step 12.2.5.2

Raise $\sqrt[3]{7}$ to the power of $1$ .

$f (10 \sqrt[3]{7}) = \frac{4200 {\sqrt[3]{7}}^{2}}{\sqrt[3]{7} {\sqrt[3]{7}}^{2}}$

Step 12.2.5.3

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

$f (10 \sqrt[3]{7}) = \frac{4200 {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1 + 2}}$

Step 12.2.5.4

Add $1$ and $2$ .

$f (10 \sqrt[3]{7}) = \frac{4200 {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{3}}$

Step 12.2.5.5

Rewrite ${\sqrt[3]{7}}^{3}$ as $7$ .

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Step 12.2.5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$f (10 \sqrt[3]{7}) = \frac{4200 {\sqrt[3]{7}}^{2}}{{(7^{\frac{1}{3}})}^{3}}$

Step 12.2.5.5.2

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

$f (10 \sqrt[3]{7}) = \frac{4200 {\sqrt[3]{7}}^{2}}{7^{\frac{1}{3} \cdot 3}}$

Step 12.2.5.5.3

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

$f (10 \sqrt[3]{7}) = \frac{4200 {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}}$

Step 12.2.5.5.4

Cancel the common factor of $3$ .

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Step 12.2.5.5.4.1

Cancel the common factor.

$f (10 \sqrt[3]{7}) = \frac{4200 {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}}$

Step 12.2.5.5.4.2

Rewrite the expression.

$f (10 \sqrt[3]{7}) = \frac{4200 {\sqrt[3]{7}}^{2}}{7}$

Step 12.2.5.5.5

Evaluate the exponent.

$f (10 \sqrt[3]{7}) = \frac{4200 {\sqrt[3]{7}}^{2}}{7}$

Step 12.2.6

Cancel the common factor of $4200$ and $7$ .

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Step 12.2.6.1

Factor $7$ out of $4200 {\sqrt[3]{7}}^{2}$ .

$f (10 \sqrt[3]{7}) = \frac{7 (600 {\sqrt[3]{7}}^{2})}{7}$

Step 12.2.6.2

Cancel the common factors.

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Step 12.2.6.2.1

Factor $7$ out of $7$ .

$f (10 \sqrt[3]{7}) = \frac{7 (600 {\sqrt[3]{7}}^{2})}{7 (1)}$

Step 12.2.6.2.2

Cancel the common factor.

$f (10 \sqrt[3]{7}) = \frac{7 (600 {\sqrt[3]{7}}^{2})}{7 \cdot 1}$

Step 12.2.6.2.3

Rewrite the expression.

$f (10 \sqrt[3]{7}) = \frac{600 {\sqrt[3]{7}}^{2}}{1}$

Step 12.2.6.2.4

Divide $600 {\sqrt[3]{7}}^{2}$ by $1$ .

$f (10 \sqrt[3]{7}) = 600 {\sqrt[3]{7}}^{2}$

Step 12.2.7

Rewrite ${\sqrt[3]{7}}^{2}$ as $\sqrt[3]{7^{2}}$ .

$f (10 \sqrt[3]{7}) = 600 \sqrt[3]{7^{2}}$

Step 12.2.8

Raise $7$ to the power of $2$ .

$f (10 \sqrt[3]{7}) = 600 \sqrt[3]{49}$

Step 12.2.9

The final answer is $600 \sqrt[3]{49}$ .

$y = 600 \sqrt[3]{49}$

Step 13

These are the local extrema for $f (x) = \frac{2 x^{3} + 28000}{x}$ .

$(10 \sqrt[3]{7}, 600 \sqrt[3]{49})$ is a local minima

Step 14