Calculus Examples

$x^{2} + \frac{480}{x}$

Step 1

Write $x^{2} + \frac{480}{x}$ as a function.

$f (x) = x^{2} + \frac{480}{x}$

Step 2

Find the first derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{2} + \frac{480}{x}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{480}{x}]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{480}{x}]$

Step 2.1.2

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

$2 x + \frac{d}{d x} [\frac{480}{x}]$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{480}{x}]$ .

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Step 2.2.1

Since $480$ is constant with respect to $x$ , the derivative of $\frac{480}{x}$ with respect to $x$ is $480 \frac{d}{d x} [\frac{1}{x}]$ .

$2 x + 480 \frac{d}{d x} [\frac{1}{x}]$

Step 2.2.2

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

$2 x + 480 \frac{d}{d x} [x^{- 1}]$

Step 2.2.3

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

$2 x + 480 (- x^{- 2})$

Step 2.2.4

Multiply $- 1$ by $480$ .

$2 x - 480 x^{- 2}$

Step 2.3

Simplify.

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Step 2.3.1

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

$2 x - 480 \frac{1}{x^{2}}$

Step 2.3.2

Combine terms.

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Step 2.3.2.1

Combine $- 480$ and $\frac{1}{x^{2}}$ .

$2 x + \frac{- 480}{x^{2}}$

Step 2.3.2.2

Move the negative in front of the fraction.

$2 x - \frac{480}{x^{2}}$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $2 x - \frac{480}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{480}{x^{2}}]$ .

$f'' (x) = \frac{d}{d x} (2 x) + \frac{d}{d x} (- \frac{480}{x^{2}})$

Step 3.2

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

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Step 3.2.1

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

$f'' (x) = 2 \frac{d}{d x} (x) + \frac{d}{d x} (- \frac{480}{x^{2}})$

Step 3.2.2

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

$f'' (x) = 2 \cdot 1 + \frac{d}{d x} (- \frac{480}{x^{2}})$

Step 3.2.3

Multiply $2$ by $1$ .

$f'' (x) = 2 + \frac{d}{d x} (- \frac{480}{x^{2}})$

Step 3.3

Evaluate $\frac{d}{d x} [- \frac{480}{x^{2}}]$ .

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Step 3.3.1

Since $- 480$ is constant with respect to $x$ , the derivative of $- \frac{480}{x^{2}}$ with respect to $x$ is $- 480 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$f'' (x) = 2 - 480 \frac{d}{d x} \frac{1}{x^{2}}$

Step 3.3.2

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

$f'' (x) = 2 - 480 \frac{d}{d x} {(x^{2})}^{- 1}$

Step 3.3.3

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 3.3.3.1

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

$f'' (x) = 2 - 480 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{2}))$

Step 3.3.3.2

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

$f'' (x) = 2 - 480 (- u^{- 2} \frac{d}{d x} x^{2})$

Step 3.3.3.3

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

$f'' (x) = 2 - 480 (- {(x^{2})}^{- 2} \frac{d}{d x} x^{2})$

Step 3.3.4

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

$f'' (x) = 2 - 480 (- {(x^{2})}^{- 2} (2 x))$

Step 3.3.5

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

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Step 3.3.5.1

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

$f'' (x) = 2 - 480 (- x^{2 \cdot - 2} (2 x))$

Step 3.3.5.2

Multiply $2$ by $- 2$ .

$f'' (x) = 2 - 480 (- x^{- 4} (2 x))$

Step 3.3.6

Multiply $2$ by $- 1$ .

$f'' (x) = 2 - 480 (- 2 x^{- 4} x)$

Step 3.3.7

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

$f'' (x) = 2 - 480 (- 2 (x \cdot x^{- 4}))$

Step 3.3.8

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

$f'' (x) = 2 - 480 (- 2 x^{1 - 4})$

Step 3.3.9

Subtract $4$ from $1$ .

$f'' (x) = 2 - 480 (- 2 x^{- 3})$

Step 3.3.10

Multiply $- 2$ by $- 480$ .

$f'' (x) = 2 + 960 x^{- 3}$

Step 3.4

Simplify.

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Step 3.4.1

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

$f'' (x) = 2 + 960 (\frac{1}{x^{3}})$

Step 3.4.2

Combine $960$ and $\frac{1}{x^{3}}$ .

$f'' (x) = 2 + \frac{960}{x^{3}}$

Step 3.4.3

Reorder terms.

$f'' (x) = \frac{960}{x^{3}} + 2$

Step 4

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

$2 x - \frac{480}{x^{2}} = 0$

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

Differentiate.

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Step 5.1.1.1

By the Sum Rule, the derivative of $x^{2} + \frac{480}{x}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{480}{x}]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{480}{x}]$

Step 5.1.1.2

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

$2 x + \frac{d}{d x} [\frac{480}{x}]$

Step 5.1.2

Evaluate $\frac{d}{d x} [\frac{480}{x}]$ .

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Step 5.1.2.1

Since $480$ is constant with respect to $x$ , the derivative of $\frac{480}{x}$ with respect to $x$ is $480 \frac{d}{d x} [\frac{1}{x}]$ .

$2 x + 480 \frac{d}{d x} [\frac{1}{x}]$

Step 5.1.2.2

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

$2 x + 480 \frac{d}{d x} [x^{- 1}]$

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 $n = - 1$ .

$2 x + 480 (- x^{- 2})$

Step 5.1.2.4

Multiply $- 1$ by $480$ .

$2 x - 480 x^{- 2}$

Step 5.1.3

Simplify.

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Step 5.1.3.1

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

$2 x - 480 \frac{1}{x^{2}}$

Step 5.1.3.2

Combine terms.

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Step 5.1.3.2.1

Combine $- 480$ and $\frac{1}{x^{2}}$ .

$2 x + \frac{- 480}{x^{2}}$

Step 5.1.3.2.2

Move the negative in front of the fraction.

$f' (x) = 2 x - \frac{480}{x^{2}}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $2 x - \frac{480}{x^{2}}$ .

$2 x - \frac{480}{x^{2}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $2 x - \frac{480}{x^{2}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$2 x - \frac{480}{x^{2}} = 0$

Step 6.2

Find the LCD of the terms in the equation.

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Step 6.2.1

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

$1, x^{2}, 1$

Step 6.2.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 6.3

Multiply each term in $2 x - \frac{480}{x^{2}} = 0$ by $x^{2}$ to eliminate the fractions.

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Step 6.3.1

Multiply each term in $2 x - \frac{480}{x^{2}} = 0$ by $x^{2}$ .

$2 x \cdot x^{2} - \frac{480}{x^{2}} x^{2} = 0 x^{2}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Simplify each term.

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Step 6.3.2.1.1

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

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Step 6.3.2.1.1.1

Move $x^{2}$ .

$2 (x^{2} x) - \frac{480}{x^{2}} x^{2} = 0 x^{2}$

Step 6.3.2.1.1.2

Multiply $x^{2}$ by $x$ .

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Step 6.3.2.1.1.2.1

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

$2 (x^{2} x^{1}) - \frac{480}{x^{2}} x^{2} = 0 x^{2}$

Step 6.3.2.1.1.2.2

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

$2 x^{2 + 1} - \frac{480}{x^{2}} x^{2} = 0 x^{2}$

Step 6.3.2.1.1.3

Add $2$ and $1$ .

$2 x^{3} - \frac{480}{x^{2}} x^{2} = 0 x^{2}$

Step 6.3.2.1.2

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

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Step 6.3.2.1.2.1

Move the leading negative in $- \frac{480}{x^{2}}$ into the numerator.

$2 x^{3} + \frac{- 480}{x^{2}} x^{2} = 0 x^{2}$

Step 6.3.2.1.2.2

Cancel the common factor.

$2 x^{3} + \frac{- 480}{x^{2}} x^{2} = 0 x^{2}$

Step 6.3.2.1.2.3

Rewrite the expression.

$2 x^{3} - 480 = 0 x^{2}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Multiply $0$ by $x^{2}$ .

$2 x^{3} - 480 = 0$

Step 6.4

Solve the equation.

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Step 6.4.1

Add $480$ to both sides of the equation.

$2 x^{3} = 480$

Step 6.4.2

Subtract $480$ from both sides of the equation.

$2 x^{3} - 480 = 0$

Step 6.4.3

Factor $2$ out of $2 x^{3} - 480$ .

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Step 6.4.3.1

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

$2 (x^{3}) - 480 = 0$

Step 6.4.3.2

Factor $2$ out of $- 480$ .

$2 x^{3} + 2 \cdot - 240 = 0$

Step 6.4.3.3

Factor $2$ out of $2 x^{3} + 2 \cdot - 240$ .

$2 (x^{3} - 240) = 0$

Step 6.4.4

Divide each term in $2 (x^{3} - 240) = 0$ by $2$ and simplify.

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Step 6.4.4.1

Divide each term in $2 (x^{3} - 240) = 0$ by $2$ .

$\frac{2 (x^{3} - 240)}{2} = \frac{0}{2}$

Step 6.4.4.2

Simplify the left side.

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Step 6.4.4.2.1

Cancel the common factor of $2$ .

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Step 6.4.4.2.1.1

Cancel the common factor.

$\frac{2 (x^{3} - 240)}{2} = \frac{0}{2}$

Step 6.4.4.2.1.2

Divide $x^{3} - 240$ by $1$ .

$x^{3} - 240 = \frac{0}{2}$

Step 6.4.4.3

Simplify the right side.

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Step 6.4.4.3.1

Divide $0$ by $2$ .

$x^{3} - 240 = 0$

Step 6.4.5

Add $240$ to both sides of the equation.

$x^{3} = 240$

Step 6.4.6

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

$x = \sqrt[3]{240}$

Step 6.4.7

Simplify $\sqrt[3]{240}$ .

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Step 6.4.7.1

Rewrite $240$ as $2^{3} \cdot 30$ .

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Step 6.4.7.1.1

Factor $8$ out of $240$ .

$x = \sqrt[3]{8 (30)}$

Step 6.4.7.1.2

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

$x = \sqrt[3]{2^{3} \cdot 30}$

Step 6.4.7.2

Pull terms out from under the radical.

$x = 2 \sqrt[3]{30}$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Set the denominator in $\frac{480}{x^{2}}$ 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.

$x = 2 \sqrt[3]{30}$

Step 9

Evaluate the second derivative at $x = 2 \sqrt[3]{30}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{960}{{(2 \sqrt[3]{30})}^{3}} + 2$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify each term.

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Step 10.1.1

Simplify the denominator.

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Step 10.1.1.1

Apply the product rule to $2 \sqrt[3]{30}$ .

$\frac{960}{2^{3} {\sqrt[3]{30}}^{3}} + 2$

Step 10.1.1.2

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

$\frac{960}{8 {\sqrt[3]{30}}^{3}} + 2$

Step 10.1.1.3

Rewrite ${\sqrt[3]{30}}^{3}$ as $30$ .

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Step 10.1.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{30}$ as $30^{\frac{1}{3}}$ .

$\frac{960}{8 {(30^{\frac{1}{3}})}^{3}} + 2$

Step 10.1.1.3.2

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

$\frac{960}{8 \cdot 30^{\frac{1}{3} \cdot 3}} + 2$

Step 10.1.1.3.3

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

$\frac{960}{8 \cdot 30^{\frac{3}{3}}} + 2$

Step 10.1.1.3.4

Cancel the common factor of $3$ .

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Step 10.1.1.3.4.1

Cancel the common factor.

$\frac{960}{8 \cdot 30^{\frac{3}{3}}} + 2$

Step 10.1.1.3.4.2

Rewrite the expression.

$\frac{960}{8 \cdot 30^{1}} + 2$

Step 10.1.1.3.5

Evaluate the exponent.

$\frac{960}{8 \cdot 30} + 2$

Step 10.1.2

Multiply $8$ by $30$ .

$\frac{960}{240} + 2$

Step 10.1.3

Divide $960$ by $240$ .

$4 + 2$

Step 10.2

Add $4$ and $2$ .

$6$

Step 11

$x = 2 \sqrt[3]{30}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 2 \sqrt[3]{30}$ is a local minimum

Step 12

Find the y-value when $x = 2 \sqrt[3]{30}$ .

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Step 12.1

Replace the variable $x$ with $2 \sqrt[3]{30}$ in the expression.

$f (2 \sqrt[3]{30}) = {(2 \sqrt[3]{30})}^{2} + \frac{480}{2 \sqrt[3]{30}}$

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

Apply the product rule to $2 \sqrt[3]{30}$ .

$f (2 \sqrt[3]{30}) = 2^{2} {\sqrt[3]{30}}^{2} + \frac{480}{2 \sqrt[3]{30}}$

Step 12.2.1.2

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

$f (2 \sqrt[3]{30}) = 4 {\sqrt[3]{30}}^{2} + \frac{480}{2 \sqrt[3]{30}}$

Step 12.2.1.3

Rewrite ${\sqrt[3]{30}}^{2}$ as $\sqrt[3]{30^{2}}$ .

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{30^{2}} + \frac{480}{2 \sqrt[3]{30}}$

Step 12.2.1.4

Raise $30$ to the power of $2$ .

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{480}{2 \sqrt[3]{30}}$

Step 12.2.1.5

Cancel the common factor of $480$ and $2$ .

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Step 12.2.1.5.1

Factor $2$ out of $480$ .

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{2 \cdot 240}{2 \sqrt[3]{30}}$

Step 12.2.1.5.2

Cancel the common factors.

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Step 12.2.1.5.2.1

Factor $2$ out of $2 \sqrt[3]{30}$ .

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{2 \cdot 240}{2 (\sqrt[3]{30})}$

Step 12.2.1.5.2.2

Cancel the common factor.

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{2 \cdot 240}{2 \sqrt[3]{30}}$

Step 12.2.1.5.2.3

Rewrite the expression.

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{240}{\sqrt[3]{30}}$

Step 12.2.1.6

Multiply $\frac{240}{\sqrt[3]{30}}$ by $\frac{{\sqrt[3]{30}}^{2}}{{\sqrt[3]{30}}^{2}}$ .

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{240}{\sqrt[3]{30}} \cdot \frac{{\sqrt[3]{30}}^{2}}{{\sqrt[3]{30}}^{2}}$

Step 12.2.1.7

Combine and simplify the denominator.

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Step 12.2.1.7.1

Multiply $\frac{240}{\sqrt[3]{30}}$ by $\frac{{\sqrt[3]{30}}^{2}}{{\sqrt[3]{30}}^{2}}$ .

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{240 {\sqrt[3]{30}}^{2}}{\sqrt[3]{30} {\sqrt[3]{30}}^{2}}$

Step 12.2.1.7.2

Raise $\sqrt[3]{30}$ to the power of $1$ .

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{240 {\sqrt[3]{30}}^{2}}{\sqrt[3]{30} {\sqrt[3]{30}}^{2}}$

Step 12.2.1.7.3

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

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{240 {\sqrt[3]{30}}^{2}}{{\sqrt[3]{30}}^{1 + 2}}$

Step 12.2.1.7.4

Add $1$ and $2$ .

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{240 {\sqrt[3]{30}}^{2}}{{\sqrt[3]{30}}^{3}}$

Step 12.2.1.7.5

Rewrite ${\sqrt[3]{30}}^{3}$ as $30$ .

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Step 12.2.1.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{30}$ as $30^{\frac{1}{3}}$ .

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{240 {\sqrt[3]{30}}^{2}}{{(30^{\frac{1}{3}})}^{3}}$

Step 12.2.1.7.5.2

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

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{240 {\sqrt[3]{30}}^{2}}{30^{\frac{1}{3} \cdot 3}}$

Step 12.2.1.7.5.3

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

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{240 {\sqrt[3]{30}}^{2}}{30^{\frac{3}{3}}}$

Step 12.2.1.7.5.4

Cancel the common factor of $3$ .

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Step 12.2.1.7.5.4.1

Cancel the common factor.

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{240 {\sqrt[3]{30}}^{2}}{30^{\frac{3}{3}}}$

Step 12.2.1.7.5.4.2

Rewrite the expression.

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{240 {\sqrt[3]{30}}^{2}}{30}$

Step 12.2.1.7.5.5

Evaluate the exponent.

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{240 {\sqrt[3]{30}}^{2}}{30}$

Step 12.2.1.8

Cancel the common factor of $240$ and $30$ .

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Step 12.2.1.8.1

Factor $30$ out of $240 {\sqrt[3]{30}}^{2}$ .

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{30 (8 {\sqrt[3]{30}}^{2})}{30}$

Step 12.2.1.8.2

Cancel the common factors.

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Step 12.2.1.8.2.1

Factor $30$ out of $30$ .

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{30 (8 {\sqrt[3]{30}}^{2})}{30 (1)}$

Step 12.2.1.8.2.2

Cancel the common factor.

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{30 (8 {\sqrt[3]{30}}^{2})}{30 \cdot 1}$

Step 12.2.1.8.2.3

Rewrite the expression.

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + \frac{8 {\sqrt[3]{30}}^{2}}{1}$

Step 12.2.1.8.2.4

Divide $8 {\sqrt[3]{30}}^{2}$ by $1$ .

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + 8 {\sqrt[3]{30}}^{2}$

Step 12.2.1.9

Rewrite ${\sqrt[3]{30}}^{2}$ as $\sqrt[3]{30^{2}}$ .

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + 8 \sqrt[3]{30^{2}}$

Step 12.2.1.10

Raise $30$ to the power of $2$ .

$f (2 \sqrt[3]{30}) = 4 \sqrt[3]{900} + 8 \sqrt[3]{900}$

Step 12.2.2

Add $4 \sqrt[3]{900}$ and $8 \sqrt[3]{900}$ .

$f (2 \sqrt[3]{30}) = 12 \sqrt[3]{900}$

Step 12.2.3

The final answer is $12 \sqrt[3]{900}$ .

$y = 12 \sqrt[3]{900}$

Step 13

These are the local extrema for $f (x) = x^{2} + \frac{480}{x}$ .

$(2 \sqrt[3]{30}, 12 \sqrt[3]{900})$ is a local minima

Step 14