Calculus Examples

$0.25 F + \frac{1.7}{F^{2}}$

Step 1

Write $0.25 F + \frac{1.7}{F^{2}}$ as a function.

$f (F) = 0.25 F + \frac{1.7}{F^{2}}$

Step 2

Find the first derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $0.25 F + \frac{1.7}{F^{2}}$ with respect to $F$ is $\frac{d}{d F} [0.25 F] + \frac{d}{d F} [\frac{1.7}{F^{2}}]$ .

$\frac{d}{d F} [0.25 F] + \frac{d}{d F} [\frac{1.7}{F^{2}}]$

Step 2.2

Evaluate $\frac{d}{d F} [0.25 F]$ .

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Step 2.2.1

Since $0.25$ is constant with respect to $F$ , the derivative of $0.25 F$ with respect to $F$ is $0.25 \frac{d}{d F} [F]$ .

$0.25 \frac{d}{d F} [F] + \frac{d}{d F} [\frac{1.7}{F^{2}}]$

Step 2.2.2

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

$0.25 \cdot 1 + \frac{d}{d F} [\frac{1.7}{F^{2}}]$

Step 2.2.3

Multiply $0.25$ by $1$ .

$0.25 + \frac{d}{d F} [\frac{1.7}{F^{2}}]$

Step 2.3

Evaluate $\frac{d}{d F} [\frac{1.7}{F^{2}}]$ .

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Step 2.3.1

Since $1.7$ is constant with respect to $F$ , the derivative of $\frac{1.7}{F^{2}}$ with respect to $F$ is $1.7 \frac{d}{d F} [\frac{1}{F^{2}}]$ .

$0.25 + 1.7 \frac{d}{d F} [\frac{1}{F^{2}}]$

Step 2.3.2

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

$0.25 + 1.7 \frac{d}{d F} [{(F^{2})}^{- 1}]$

Step 2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d F} [f (g (F))]$ is $f' (g (F)) g' (F)$ where $f (F) = F^{- 1}$ and $g (F) = F^{2}$ .

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Step 2.3.3.1

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

$0.25 + 1.7 (\frac{d}{d u} [u^{- 1}] \frac{d}{d F} [F^{2}])$

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 $n = - 1$ .

$0.25 + 1.7 (- u^{- 2} \frac{d}{d F} [F^{2}])$

Step 2.3.3.3

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

$0.25 + 1.7 (- {(F^{2})}^{- 2} \frac{d}{d F} [F^{2}])$

Step 2.3.4

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

$0.25 + 1.7 (- {(F^{2})}^{- 2} (2 F))$

Step 2.3.5

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

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Step 2.3.5.1

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

$0.25 + 1.7 (- F^{2 \cdot - 2} (2 F))$

Step 2.3.5.2

Multiply $2$ by $- 2$ .

$0.25 + 1.7 (- F^{- 4} (2 F))$

Step 2.3.6

Multiply $2$ by $- 1$ .

$0.25 + 1.7 (- 2 F^{- 4} F)$

Step 2.3.7

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

$0.25 + 1.7 (- 2 (F^{1} F^{- 4}))$

Step 2.3.8

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

$0.25 + 1.7 (- 2 F^{1 - 4})$

Step 2.3.9

Subtract $4$ from $1$ .

$0.25 + 1.7 (- 2 F^{- 3})$

Step 2.3.10

Multiply $- 2$ by $1.7$ .

$0.25 - 3.4 F^{- 3}$

Step 2.4

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

$0.25 - 3.4 \frac{1}{F^{3}}$

Step 2.5

Simplify.

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Step 2.5.1

Combine terms.

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Step 2.5.1.1

Combine $- 3.4$ and $\frac{1}{F^{3}}$ .

$0.25 + \frac{- 3.4}{F^{3}}$

Step 2.5.1.2

Move the negative in front of the fraction.

$0.25 - \frac{3.4}{F^{3}}$

Step 2.5.2

Reorder terms.

$- \frac{3.4}{F^{3}} + 0.25$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $- \frac{3.4}{F^{3}} + 0.25$ with respect to $F$ is $\frac{d}{d F} [- \frac{3.4}{F^{3}}] + \frac{d}{d F} [0.25]$ .

$f'' (F) = \frac{d}{d F} (- \frac{3.4}{F^{3}}) + \frac{d}{d F} (0.25)$

Step 3.2

Evaluate $\frac{d}{d F} [- \frac{3.4}{F^{3}}]$ .

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Step 3.2.1

Since $- 3.4$ is constant with respect to $F$ , the derivative of $- \frac{3.4}{F^{3}}$ with respect to $F$ is $- 3.4 \frac{d}{d F} [\frac{1}{F^{3}}]$ .

$f'' (F) = - 3.4 \frac{d}{d F} \frac{1}{F^{3}} + \frac{d}{d F} (0.25)$

Step 3.2.2

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

$f'' (F) = - 3.4 \frac{d}{d F} {(F^{3})}^{- 1} + \frac{d}{d F} (0.25)$

Step 3.2.3

Differentiate using the chain rule, which states that $\frac{d}{d F} [f (g (F))]$ is $f' (g (F)) g' (F)$ where $f (F) = F^{- 1}$ and $g (F) = F^{3}$ .

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Step 3.2.3.1

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

$f'' (F) = - 3.4 (\frac{d}{d u} (u^{- 1}) \frac{d}{d F} (F^{3})) + \frac{d}{d F} (0.25)$

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 $n = - 1$ .

$f'' (F) = - 3.4 (- u^{- 2} \frac{d}{d F} F^{3}) + \frac{d}{d F} (0.25)$

Step 3.2.3.3

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

$f'' (F) = - 3.4 (- {(F^{3})}^{- 2} \frac{d}{d F} F^{3}) + \frac{d}{d F} (0.25)$

Step 3.2.4

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

$f'' (F) = - 3.4 (- {(F^{3})}^{- 2} (3 F^{2})) + \frac{d}{d F} (0.25)$

Step 3.2.5

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

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Step 3.2.5.1

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

$f'' (F) = - 3.4 (- F^{3 \cdot - 2} (3 F^{2})) + \frac{d}{d F} (0.25)$

Step 3.2.5.2

Multiply $3$ by $- 2$ .

$f'' (F) = - 3.4 (- F^{- 6} (3 F^{2})) + \frac{d}{d F} (0.25)$

Step 3.2.6

Multiply $3$ by $- 1$ .

$f'' (F) = - 3.4 (- 3 F^{- 6} F^{2}) + \frac{d}{d F} (0.25)$

Step 3.2.7

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

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Step 3.2.7.1

Move $F^{2}$ .

$f'' (F) = - 3.4 (- 3 (F^{2} F^{- 6})) + \frac{d}{d F} (0.25)$

Step 3.2.7.2

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

$f'' (F) = - 3.4 (- 3 F^{2 - 6}) + \frac{d}{d F} (0.25)$

Step 3.2.7.3

Subtract $6$ from $2$ .

$f'' (F) = - 3.4 (- 3 F^{- 4}) + \frac{d}{d F} (0.25)$

Step 3.2.8

Multiply $- 3$ by $- 3.4$ .

$f'' (F) = 10.2 F^{- 4} + \frac{d}{d F} (0.25)$

Step 3.3

Since $0.25$ is constant with respect to $F$ , the derivative of $0.25$ with respect to $F$ is $0$ .

$f'' (F) = 10.2 F^{- 4} + 0$

Step 3.4

Simplify.

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Step 3.4.1

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

$f'' (F) = 10.2 (\frac{1}{F^{4}}) + 0$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Combine $10.2$ and $\frac{1}{F^{4}}$ .

$f'' (F) = \frac{10.2}{F^{4}} + 0$

Step 3.4.2.2

Add $\frac{10.2}{F^{4}}$ and $0$ .

$f'' (F) = \frac{10.2}{F^{4}}$

Step 4

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

$- \frac{3.4}{F^{3}} + 0.25 = 0$

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 $0.25 F + \frac{1.7}{F^{2}}$ with respect to $F$ is $\frac{d}{d F} [0.25 F] + \frac{d}{d F} [\frac{1.7}{F^{2}}]$ .

$\frac{d}{d F} [0.25 F] + \frac{d}{d F} [\frac{1.7}{F^{2}}]$

Step 5.1.2

Evaluate $\frac{d}{d F} [0.25 F]$ .

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Step 5.1.2.1

Since $0.25$ is constant with respect to $F$ , the derivative of $0.25 F$ with respect to $F$ is $0.25 \frac{d}{d F} [F]$ .

$0.25 \frac{d}{d F} [F] + \frac{d}{d F} [\frac{1.7}{F^{2}}]$

Step 5.1.2.2

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

$0.25 \cdot 1 + \frac{d}{d F} [\frac{1.7}{F^{2}}]$

Step 5.1.2.3

Multiply $0.25$ by $1$ .

$0.25 + \frac{d}{d F} [\frac{1.7}{F^{2}}]$

Step 5.1.3

Evaluate $\frac{d}{d F} [\frac{1.7}{F^{2}}]$ .

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Step 5.1.3.1

Since $1.7$ is constant with respect to $F$ , the derivative of $\frac{1.7}{F^{2}}$ with respect to $F$ is $1.7 \frac{d}{d F} [\frac{1}{F^{2}}]$ .

$0.25 + 1.7 \frac{d}{d F} [\frac{1}{F^{2}}]$

Step 5.1.3.2

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

$0.25 + 1.7 \frac{d}{d F} [{(F^{2})}^{- 1}]$

Step 5.1.3.3

Differentiate using the chain rule, which states that $\frac{d}{d F} [f (g (F))]$ is $f' (g (F)) g' (F)$ where $f (F) = F^{- 1}$ and $g (F) = F^{2}$ .

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Step 5.1.3.3.1

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

$0.25 + 1.7 (\frac{d}{d u} [u^{- 1}] \frac{d}{d F} [F^{2}])$

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 $n = - 1$ .

$0.25 + 1.7 (- u^{- 2} \frac{d}{d F} [F^{2}])$

Step 5.1.3.3.3

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

$0.25 + 1.7 (- {(F^{2})}^{- 2} \frac{d}{d F} [F^{2}])$

Step 5.1.3.4

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

$0.25 + 1.7 (- {(F^{2})}^{- 2} (2 F))$

Step 5.1.3.5

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

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Step 5.1.3.5.1

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

$0.25 + 1.7 (- F^{2 \cdot - 2} (2 F))$

Step 5.1.3.5.2

Multiply $2$ by $- 2$ .

$0.25 + 1.7 (- F^{- 4} (2 F))$

Step 5.1.3.6

Multiply $2$ by $- 1$ .

$0.25 + 1.7 (- 2 F^{- 4} F)$

Step 5.1.3.7

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

$0.25 + 1.7 (- 2 (F^{1} F^{- 4}))$

Step 5.1.3.8

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

$0.25 + 1.7 (- 2 F^{1 - 4})$

Step 5.1.3.9

Subtract $4$ from $1$ .

$0.25 + 1.7 (- 2 F^{- 3})$

Step 5.1.3.10

Multiply $- 2$ by $1.7$ .

$0.25 - 3.4 F^{- 3}$

Step 5.1.4

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

$0.25 - 3.4 \frac{1}{F^{3}}$

Step 5.1.5

Simplify.

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Step 5.1.5.1

Combine terms.

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Step 5.1.5.1.1

Combine $- 3.4$ and $\frac{1}{F^{3}}$ .

$0.25 + \frac{- 3.4}{F^{3}}$

Step 5.1.5.1.2

Move the negative in front of the fraction.

$0.25 - \frac{3.4}{F^{3}}$

Step 5.1.5.2

Reorder terms.

$f' (F) = - \frac{3.4}{F^{3}} + 0.25$

Step 5.2

The first derivative of $f (F)$ with respect to $F$ is $- \frac{3.4}{F^{3}} + 0.25$ .

$- \frac{3.4}{F^{3}} + 0.25$

Step 6

Set the first derivative equal to $0$ then solve the equation $- \frac{3.4}{F^{3}} + 0.25 = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$- \frac{3.4}{F^{3}} + 0.25 = 0$

Step 6.2

Subtract $0.25$ from both sides of the equation.

$- \frac{3.4}{F^{3}} = - 0.25$

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.

$F^{3}, 1$

Step 6.3.2

The LCM of one and any expression is the expression.

$F^{3}$

Step 6.4

Multiply each term in $- \frac{3.4}{F^{3}} = - 0.25$ by $F^{3}$ to eliminate the fractions.

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Step 6.4.1

Multiply each term in $- \frac{3.4}{F^{3}} = - 0.25$ by $F^{3}$ .

$- \frac{3.4}{F^{3}} F^{3} = - 0.25 F^{3}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

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

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Step 6.4.2.1.1

Move the leading negative in $- \frac{3.4}{F^{3}}$ into the numerator.

$\frac{- 3.4}{F^{3}} F^{3} = - 0.25 F^{3}$

Step 6.4.2.1.2

Cancel the common factor.

$\frac{- 3.4}{F^{3}} F^{3} = - 0.25 F^{3}$

Step 6.4.2.1.3

Rewrite the expression.

$- 3.4 = - 0.25 F^{3}$

Step 6.5

Solve the equation.

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Step 6.5.1

Rewrite the equation as $- 0.25 F^{3} = - 3.4$ .

$- 0.25 F^{3} = - 3.4$

Step 6.5.2

Add $3.4$ to both sides of the equation.

$- 0.25 F^{3} + 3.4 = 0$

Step 6.5.3

Factor $- 0.05$ out of $- 0.25 F^{3} + 3.4$ .

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Step 6.5.3.1

Factor $- 0.05$ out of $- 0.25 F^{3}$ .

$- 0.05 (5 F^{3}) + 3.4 = 0$

Step 6.5.3.2

Factor $- 0.05$ out of $3.4$ .

$- 0.05 (5 F^{3}) - 0.05 \cdot - 68 = 0$

Step 6.5.3.3

Factor $- 0.05$ out of $- 0.05 (5 F^{3}) - 0.05 (- 68)$ .

$- 0.05 (5 F^{3} - 68) = 0$

Step 6.5.4

Divide each term in $- 0.05 (5 F^{3} - 68) = 0$ by $- 0.05$ and simplify.

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Step 6.5.4.1

Divide each term in $- 0.05 (5 F^{3} - 68) = 0$ by $- 0.05$ .

$\frac{- 0.05 (5 F^{3} - 68)}{- 0.05} = \frac{0}{- 0.05}$

Step 6.5.4.2

Simplify the left side.

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Step 6.5.4.2.1

Cancel the common factor of $- 0.05$ .

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Step 6.5.4.2.1.1

Cancel the common factor.

$\frac{- 0.05 (5 F^{3} - 68)}{- 0.05} = \frac{0}{- 0.05}$

Step 6.5.4.2.1.2

Divide $5 F^{3} - 68$ by $1$ .

$5 F^{3} - 68 = \frac{0}{- 0.05}$

Step 6.5.4.3

Simplify the right side.

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Step 6.5.4.3.1

Divide $0$ by $- 0.05$ .

$5 F^{3} - 68 = 0$

Step 6.5.5

Add $68$ to both sides of the equation.

$5 F^{3} = 68$

Step 6.5.6

Divide each term in $5 F^{3} = 68$ by $5$ and simplify.

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Step 6.5.6.1

Divide each term in $5 F^{3} = 68$ by $5$ .

$\frac{5 F^{3}}{5} = \frac{68}{5}$

Step 6.5.6.2

Simplify the left side.

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Step 6.5.6.2.1

Cancel the common factor of $5$ .

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Step 6.5.6.2.1.1

Cancel the common factor.

$\frac{5 F^{3}}{5} = \frac{68}{5}$

Step 6.5.6.2.1.2

Divide $F^{3}$ by $1$ .

$F^{3} = \frac{68}{5}$

Step 6.5.7

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

$F = \sqrt[3]{\frac{68}{5}}$

Step 6.5.8

Simplify $\sqrt[3]{\frac{68}{5}}$ .

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Step 6.5.8.1

Rewrite $\sqrt[3]{\frac{68}{5}}$ as $\frac{\sqrt[3]{68}}{\sqrt[3]{5}}$ .

$F = \frac{\sqrt[3]{68}}{\sqrt[3]{5}}$

Step 6.5.8.2

Multiply $\frac{\sqrt[3]{68}}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$F = \frac{\sqrt[3]{68}}{\sqrt[3]{5}} \cdot \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$

Step 6.5.8.3

Combine and simplify the denominator.

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Step 6.5.8.3.1

Multiply $\frac{\sqrt[3]{68}}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$F = \frac{\sqrt[3]{68} {\sqrt[3]{5}}^{2}}{\sqrt[3]{5} {\sqrt[3]{5}}^{2}}$

Step 6.5.8.3.2

Raise $\sqrt[3]{5}$ to the power of $1$ .

$F = \frac{\sqrt[3]{68} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1} {\sqrt[3]{5}}^{2}}$

Step 6.5.8.3.3

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

$F = \frac{\sqrt[3]{68} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1 + 2}}$

Step 6.5.8.3.4

Add $1$ and $2$ .

$F = \frac{\sqrt[3]{68} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{3}}$

Step 6.5.8.3.5

Rewrite ${\sqrt[3]{5}}^{3}$ as $5$ .

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Step 6.5.8.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$F = \frac{\sqrt[3]{68} {\sqrt[3]{5}}^{2}}{{(5^{\frac{1}{3}})}^{3}}$

Step 6.5.8.3.5.2

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

$F = \frac{\sqrt[3]{68} {\sqrt[3]{5}}^{2}}{5^{\frac{1}{3} \cdot 3}}$

Step 6.5.8.3.5.3

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

$F = \frac{\sqrt[3]{68} {\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

Step 6.5.8.3.5.4

Cancel the common factor of $3$ .

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Step 6.5.8.3.5.4.1

Cancel the common factor.

$F = \frac{\sqrt[3]{68} {\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

Step 6.5.8.3.5.4.2

Rewrite the expression.

$F = \frac{\sqrt[3]{68} {\sqrt[3]{5}}^{2}}{5^{1}}$

Step 6.5.8.3.5.5

Evaluate the exponent.

$F = \frac{\sqrt[3]{68} {\sqrt[3]{5}}^{2}}{5}$

Step 6.5.8.4

Simplify the numerator.

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Step 6.5.8.4.1

Rewrite ${\sqrt[3]{5}}^{2}$ as $\sqrt[3]{5^{2}}$ .

$F = \frac{\sqrt[3]{68} \sqrt[3]{5^{2}}}{5}$

Step 6.5.8.4.2

Raise $5$ to the power of $2$ .

$F = \frac{\sqrt[3]{68} \sqrt[3]{25}}{5}$

Step 6.5.8.5

Simplify the numerator.

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Step 6.5.8.5.1

Combine using the product rule for radicals.

$F = \frac{\sqrt[3]{68 \cdot 25}}{5}$

Step 6.5.8.5.2

Multiply $68$ by $25$ .

$F = \frac{\sqrt[3]{1700}}{5}$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Set the denominator in $\frac{3.4}{F^{3}}$ equal to $0$ to find where the expression is undefined.

$F^{3} = 0$

Step 7.2

Solve for $F$ .

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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.

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

Step 7.2.2

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

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Step 7.2.2.1

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

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

Step 7.2.2.2

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

$F = 0$

Step 8

Critical points to evaluate.

$F = \frac{\sqrt[3]{1700}}{5}$

Step 9

Evaluate the second derivative at $F = \frac{\sqrt[3]{1700}}{5}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{10.2}{{(\frac{\sqrt[3]{1700}}{5})}^{4}}$

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 $\frac{\sqrt[3]{1700}}{5}$ .

$\frac{10.2}{\frac{{\sqrt[3]{1700}}^{4}}{5^{4}}}$

Step 10.1.2

Simplify the numerator.

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Step 10.1.2.1

Rewrite ${\sqrt[3]{1700}}^{4}$ as $\sqrt[3]{1700^{4}}$ .

$\frac{10.2}{\frac{\sqrt[3]{1700^{4}}}{5^{4}}}$

Step 10.1.2.2

Raise $1700$ to the power of $4$ .

$\frac{10.2}{\frac{\sqrt[3]{8352100000000}}{5^{4}}}$

Step 10.1.2.3

Rewrite $8352100000000$ as $1700^{3} \cdot 1700$ .

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Step 10.1.2.3.1

Factor $4913000000$ out of $8352100000000$ .

$\frac{10.2}{\frac{\sqrt[3]{4913000000 (1700)}}{5^{4}}}$

Step 10.1.2.3.2

Rewrite $4913000000$ as $1700^{3}$ .

$\frac{10.2}{\frac{\sqrt[3]{1700^{3} \cdot 1700}}{5^{4}}}$

Step 10.1.2.4

Pull terms out from under the radical.

$\frac{10.2}{\frac{1700 \sqrt[3]{1700}}{5^{4}}}$

Step 10.1.3

Raise $5$ to the power of $4$ .

$\frac{10.2}{\frac{1700 \sqrt[3]{1700}}{625}}$

Step 10.1.4

Cancel the common factor of $1700$ and $625$ .

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Step 10.1.4.1

Factor $25$ out of $1700 \sqrt[3]{1700}$ .

$\frac{10.2}{\frac{25 (68 \sqrt[3]{1700})}{625}}$

Step 10.1.4.2

Cancel the common factors.

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Step 10.1.4.2.1

Factor $25$ out of $625$ .

$\frac{10.2}{\frac{25 (68 \sqrt[3]{1700})}{25 (25)}}$

Step 10.1.4.2.2

Cancel the common factor.

$\frac{10.2}{\frac{25 (68 \sqrt[3]{1700})}{25 \cdot 25}}$

Step 10.1.4.2.3

Rewrite the expression.

$\frac{10.2}{\frac{68 \sqrt[3]{1700}}{25}}$

Step 10.2

Multiply the numerator by the reciprocal of the denominator.

$10.2 \frac{25}{68 \sqrt[3]{1700}}$

Step 10.3

Multiply $\frac{25}{68 \sqrt[3]{1700}}$ by $\frac{{\sqrt[3]{1700}}^{2}}{{\sqrt[3]{1700}}^{2}}$ .

$10.2 (\frac{25}{68 \sqrt[3]{1700}} \cdot \frac{{\sqrt[3]{1700}}^{2}}{{\sqrt[3]{1700}}^{2}})$

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 $\frac{25}{68 \sqrt[3]{1700}}$ by $\frac{{\sqrt[3]{1700}}^{2}}{{\sqrt[3]{1700}}^{2}}$ .

$10.2 \frac{25 {\sqrt[3]{1700}}^{2}}{68 \sqrt[3]{1700} {\sqrt[3]{1700}}^{2}}$

Step 10.4.1.2

Move $\sqrt[3]{1700}$ .

$10.2 \frac{25 {\sqrt[3]{1700}}^{2}}{68 (\sqrt[3]{1700} {\sqrt[3]{1700}}^{2})}$

Step 10.4.1.3

Raise $\sqrt[3]{1700}$ to the power of $1$ .

$10.2 \frac{25 {\sqrt[3]{1700}}^{2}}{68 ({\sqrt[3]{1700}}^{1} {\sqrt[3]{1700}}^{2})}$

Step 10.4.1.4

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

$10.2 \frac{25 {\sqrt[3]{1700}}^{2}}{68 {\sqrt[3]{1700}}^{1 + 2}}$

Step 10.4.1.5

Add $1$ and $2$ .

$10.2 \frac{25 {\sqrt[3]{1700}}^{2}}{68 {\sqrt[3]{1700}}^{3}}$

Step 10.4.1.6

Rewrite ${\sqrt[3]{1700}}^{3}$ as $1700$ .

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Step 10.4.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{1700}$ as $1700^{\frac{1}{3}}$ .

$10.2 \frac{25 {\sqrt[3]{1700}}^{2}}{68 {(1700^{\frac{1}{3}})}^{3}}$

Step 10.4.1.6.2

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

$10.2 \frac{25 {\sqrt[3]{1700}}^{2}}{68 \cdot 1700^{\frac{1}{3} \cdot 3}}$

Step 10.4.1.6.3

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

$10.2 \frac{25 {\sqrt[3]{1700}}^{2}}{68 \cdot 1700^{\frac{3}{3}}}$

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.

$10.2 \frac{25 {\sqrt[3]{1700}}^{2}}{68 \cdot 1700^{\frac{3}{3}}}$

Step 10.4.1.6.4.2

Rewrite the expression.

$10.2 \frac{25 {\sqrt[3]{1700}}^{2}}{68 \cdot 1700^{1}}$

Step 10.4.1.6.5

Evaluate the exponent.

$10.2 \frac{25 {\sqrt[3]{1700}}^{2}}{68 \cdot 1700}$

Step 10.4.2

Cancel the common factor of $25$ and $1700$ .

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Step 10.4.2.1

Factor $25$ out of $25 {\sqrt[3]{1700}}^{2}$ .

$10.2 \frac{25 ({\sqrt[3]{1700}}^{2})}{68 \cdot 1700}$

Step 10.4.2.2

Cancel the common factors.

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Step 10.4.2.2.1

Factor $25$ out of $68 \cdot 1700$ .

$10.2 \frac{25 ({\sqrt[3]{1700}}^{2})}{25 (68 \cdot 68)}$

Step 10.4.2.2.2

Cancel the common factor.

$10.2 \frac{25 {\sqrt[3]{1700}}^{2}}{25 (68 \cdot 68)}$

Step 10.4.2.2.3

Rewrite the expression.

$10.2 \frac{{\sqrt[3]{1700}}^{2}}{68 \cdot 68}$

Step 10.5

Simplify the numerator.

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Step 10.5.1

Rewrite ${\sqrt[3]{1700}}^{2}$ as $\sqrt[3]{1700^{2}}$ .

$10.2 \frac{\sqrt[3]{1700^{2}}}{68 \cdot 68}$

Step 10.5.2

Raise $1700$ to the power of $2$ .

$10.2 \frac{\sqrt[3]{2890000}}{68 \cdot 68}$

Step 10.5.3

Rewrite $2890000$ as $10^{3} \cdot 2890$ .

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Step 10.5.3.1

Factor $1000$ out of $2890000$ .

$10.2 \frac{\sqrt[3]{1000 (2890)}}{68 \cdot 68}$

Step 10.5.3.2

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

$10.2 \frac{\sqrt[3]{10^{3} \cdot 2890}}{68 \cdot 68}$

Step 10.5.4

Pull terms out from under the radical.

$10.2 \frac{10 \sqrt[3]{2890}}{68 \cdot 68}$

Step 10.6

Reduce the expression by cancelling the common factors.

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Step 10.6.1

Multiply $68$ by $68$ .

$10.2 \frac{10 \sqrt[3]{2890}}{4624}$

Step 10.6.2

Cancel the common factor of $10$ and $4624$ .

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Step 10.6.2.1

Factor $2$ out of $10 \sqrt[3]{2890}$ .

$10.2 \frac{2 (5 \sqrt[3]{2890})}{4624}$

Step 10.6.2.2

Cancel the common factors.

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Step 10.6.2.2.1

Factor $2$ out of $4624$ .

$10.2 \frac{2 (5 \sqrt[3]{2890})}{2 (2312)}$

Step 10.6.2.2.2

Cancel the common factor.

$10.2 \frac{2 (5 \sqrt[3]{2890})}{2 \cdot 2312}$

Step 10.6.2.2.3

Rewrite the expression.

$10.2 \frac{5 \sqrt[3]{2890}}{2312}$

Step 10.7

Multiply $10.2 \frac{5 \sqrt[3]{2890}}{2312}$ .

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Step 10.7.1

Combine $10.2$ and $\frac{5 \sqrt[3]{2890}}{2312}$ .

$\frac{10.2 (5 \sqrt[3]{2890})}{2312}$

Step 10.7.2

Multiply $5$ by $10.2$ .

$\frac{51 \sqrt[3]{2890}}{2312}$

Step 10.7.3

Multiply $51$ by $\sqrt[3]{2890}$ .

$\frac{726.445086}{2312}$

Step 10.8

Divide $726.445086$ by $2312$ .

$0.31420635$

Step 11

$F = \frac{\sqrt[3]{1700}}{5}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$F = \frac{\sqrt[3]{1700}}{5}$ is a local minimum

Step 12

Find the y-value when $F = \frac{\sqrt[3]{1700}}{5}$ .

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Step 12.1

Replace the variable $F$ with $\frac{\sqrt[3]{1700}}{5}$ in the expression.

$f (\frac{\sqrt[3]{1700}}{5}) = 0.25 (\frac{\sqrt[3]{1700}}{5}) + \frac{1.7}{{(\frac{\sqrt[3]{1700}}{5})}^{2}}$

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

Cancel the common factor of $0.25$ .

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Step 12.2.1.1.1

Factor $0.25$ out of $5$ .

$f (\frac{\sqrt[3]{1700}}{5}) = 0.25 (\frac{\sqrt[3]{1700}}{0.25 (20)}) + \frac{1.7}{{(\frac{\sqrt[3]{1700}}{5})}^{2}}$

Step 12.2.1.1.2

Cancel the common factor.

$f (\frac{\sqrt[3]{1700}}{5}) = 0.25 (\frac{\sqrt[3]{1700}}{0.25 \cdot 20}) + \frac{1.7}{{(\frac{\sqrt[3]{1700}}{5})}^{2}}$

Step 12.2.1.1.3

Rewrite the expression.

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + \frac{1.7}{{(\frac{\sqrt[3]{1700}}{5})}^{2}}$

Step 12.2.1.2

Simplify the denominator.

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Step 12.2.1.2.1

Apply the product rule to $\frac{\sqrt[3]{1700}}{5}$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + \frac{1.7}{\frac{{\sqrt[3]{1700}}^{2}}{5^{2}}}$

Step 12.2.1.2.2

Simplify the numerator.

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Step 12.2.1.2.2.1

Rewrite ${\sqrt[3]{1700}}^{2}$ as $\sqrt[3]{1700^{2}}$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + \frac{1.7}{\frac{\sqrt[3]{1700^{2}}}{5^{2}}}$

Step 12.2.1.2.2.2

Raise $1700$ to the power of $2$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + \frac{1.7}{\frac{\sqrt[3]{2890000}}{5^{2}}}$

Step 12.2.1.2.2.3

Rewrite $2890000$ as $10^{3} \cdot 2890$ .

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Step 12.2.1.2.2.3.1

Factor $1000$ out of $2890000$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + \frac{1.7}{\frac{\sqrt[3]{1000 (2890)}}{5^{2}}}$

Step 12.2.1.2.2.3.2

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

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + \frac{1.7}{\frac{\sqrt[3]{10^{3} \cdot 2890}}{5^{2}}}$

Step 12.2.1.2.2.4

Pull terms out from under the radical.

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + \frac{1.7}{\frac{10 \sqrt[3]{2890}}{5^{2}}}$

Step 12.2.1.2.3

Raise $5$ to the power of $2$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + \frac{1.7}{\frac{10 \sqrt[3]{2890}}{25}}$

Step 12.2.1.2.4

Cancel the common factor of $10$ and $25$ .

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Step 12.2.1.2.4.1

Factor $5$ out of $10 \sqrt[3]{2890}$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + \frac{1.7}{\frac{5 (2 \sqrt[3]{2890})}{25}}$

Step 12.2.1.2.4.2

Cancel the common factors.

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Step 12.2.1.2.4.2.1

Factor $5$ out of $25$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + \frac{1.7}{\frac{5 (2 \sqrt[3]{2890})}{5 (5)}}$

Step 12.2.1.2.4.2.2

Cancel the common factor.

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + \frac{1.7}{\frac{5 (2 \sqrt[3]{2890})}{5 \cdot 5}}$

Step 12.2.1.2.4.2.3

Rewrite the expression.

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + \frac{1.7}{\frac{2 \sqrt[3]{2890}}{5}}$

Step 12.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{5}{2 \sqrt[3]{2890}})$

Step 12.2.1.4

Multiply $\frac{5}{2 \sqrt[3]{2890}}$ by $\frac{{\sqrt[3]{2890}}^{2}}{{\sqrt[3]{2890}}^{2}}$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{5}{2 \sqrt[3]{2890}} \cdot \frac{{\sqrt[3]{2890}}^{2}}{{\sqrt[3]{2890}}^{2}})$

Step 12.2.1.5

Combine and simplify the denominator.

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Step 12.2.1.5.1

Multiply $\frac{5}{2 \sqrt[3]{2890}}$ by $\frac{{\sqrt[3]{2890}}^{2}}{{\sqrt[3]{2890}}^{2}}$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{5 {\sqrt[3]{2890}}^{2}}{2 \sqrt[3]{2890} {\sqrt[3]{2890}}^{2}})$

Step 12.2.1.5.2

Move $\sqrt[3]{2890}$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{5 {\sqrt[3]{2890}}^{2}}{2 (\sqrt[3]{2890} {\sqrt[3]{2890}}^{2})})$

Step 12.2.1.5.3

Raise $\sqrt[3]{2890}$ to the power of $1$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{5 {\sqrt[3]{2890}}^{2}}{2 (\sqrt[3]{2890} {\sqrt[3]{2890}}^{2})})$

Step 12.2.1.5.4

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

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{5 {\sqrt[3]{2890}}^{2}}{2 {\sqrt[3]{2890}}^{1 + 2}})$

Step 12.2.1.5.5

Add $1$ and $2$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{5 {\sqrt[3]{2890}}^{2}}{2 {\sqrt[3]{2890}}^{3}})$

Step 12.2.1.5.6

Rewrite ${\sqrt[3]{2890}}^{3}$ as $2890$ .

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Step 12.2.1.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2890}$ as $2890^{\frac{1}{3}}$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{5 {\sqrt[3]{2890}}^{2}}{2 {(2890^{\frac{1}{3}})}^{3}})$

Step 12.2.1.5.6.2

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

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{5 {\sqrt[3]{2890}}^{2}}{2 \cdot 2890^{\frac{1}{3} \cdot 3}})$

Step 12.2.1.5.6.3

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

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{5 {\sqrt[3]{2890}}^{2}}{2 \cdot 2890^{\frac{3}{3}}})$

Step 12.2.1.5.6.4

Cancel the common factor of $3$ .

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Step 12.2.1.5.6.4.1

Cancel the common factor.

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{5 {\sqrt[3]{2890}}^{2}}{2 \cdot 2890^{\frac{3}{3}}})$

Step 12.2.1.5.6.4.2

Rewrite the expression.

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{5 {\sqrt[3]{2890}}^{2}}{2 \cdot 2890})$

Step 12.2.1.5.6.5

Evaluate the exponent.

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{5 {\sqrt[3]{2890}}^{2}}{2 \cdot 2890})$

Step 12.2.1.6

Cancel the common factor of $5$ and $2890$ .

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Step 12.2.1.6.1

Factor $5$ out of $5 {\sqrt[3]{2890}}^{2}$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{5 ({\sqrt[3]{2890}}^{2})}{2 \cdot 2890})$

Step 12.2.1.6.2

Cancel the common factors.

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Step 12.2.1.6.2.1

Factor $5$ out of $2 \cdot 2890$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{5 ({\sqrt[3]{2890}}^{2})}{5 (2 \cdot 578)})$

Step 12.2.1.6.2.2

Cancel the common factor.

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{5 {\sqrt[3]{2890}}^{2}}{5 (2 \cdot 578)})$

Step 12.2.1.6.2.3

Rewrite the expression.

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{{\sqrt[3]{2890}}^{2}}{2 \cdot 578})$

Step 12.2.1.7

Simplify the numerator.

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Step 12.2.1.7.1

Rewrite ${\sqrt[3]{2890}}^{2}$ as $\sqrt[3]{2890^{2}}$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{\sqrt[3]{2890^{2}}}{2 \cdot 578})$

Step 12.2.1.7.2

Raise $2890$ to the power of $2$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{\sqrt[3]{8352100}}{2 \cdot 578})$

Step 12.2.1.7.3

Rewrite $8352100$ as $17^{3} \cdot 1700$ .

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Step 12.2.1.7.3.1

Factor $4913$ out of $8352100$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{\sqrt[3]{4913 (1700)}}{2 \cdot 578})$

Step 12.2.1.7.3.2

Rewrite $4913$ as $17^{3}$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{\sqrt[3]{17^{3} \cdot 1700}}{2 \cdot 578})$

Step 12.2.1.7.4

Pull terms out from under the radical.

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{17 \sqrt[3]{1700}}{2 \cdot 578})$

Step 12.2.1.8

Multiply $2$ by $578$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{17 \sqrt[3]{1700}}{1156})$

Step 12.2.1.9

Cancel the common factor of $1.7$ .

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Step 12.2.1.9.1

Factor $1.7$ out of $1156$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{17 \sqrt[3]{1700}}{1.7 (680)})$

Step 12.2.1.9.2

Cancel the common factor.

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + 1.7 (\frac{17 \sqrt[3]{1700}}{1.7 \cdot 680})$

Step 12.2.1.9.3

Rewrite the expression.

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + \frac{17 \sqrt[3]{1700}}{680}$

Step 12.2.1.10

Cancel the common factor of $17$ and $680$ .

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Step 12.2.1.10.1

Factor $17$ out of $17 \sqrt[3]{1700}$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + \frac{17 (\sqrt[3]{1700})}{680}$

Step 12.2.1.10.2

Cancel the common factors.

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Step 12.2.1.10.2.1

Factor $17$ out of $680$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + \frac{17 \sqrt[3]{1700}}{17 \cdot 40}$

Step 12.2.1.10.2.2

Cancel the common factor.

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + \frac{17 \sqrt[3]{1700}}{17 \cdot 40}$

Step 12.2.1.10.2.3

Rewrite the expression.

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} + \frac{\sqrt[3]{1700}}{40}$

Step 12.2.2

To write $\frac{\sqrt[3]{1700}}{20}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700}}{20} \cdot \frac{2}{2} + \frac{\sqrt[3]{1700}}{40}$

Step 12.2.3

Write each expression with a common denominator of $40$ , by multiplying each by an appropriate factor of $1$ .

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Step 12.2.3.1

Multiply $\frac{\sqrt[3]{1700}}{20}$ by $\frac{2}{2}$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700} \cdot 2}{20 \cdot 2} + \frac{\sqrt[3]{1700}}{40}$

Step 12.2.3.2

Multiply $20$ by $2$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700} \cdot 2}{40} + \frac{\sqrt[3]{1700}}{40}$

Step 12.2.4

Combine the numerators over the common denominator.

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{\sqrt[3]{1700} \cdot 2 + \sqrt[3]{1700}}{40}$

Step 12.2.5

Simplify the numerator.

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Step 12.2.5.1

Move $2$ to the left of $\sqrt[3]{1700}$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{2 \cdot \sqrt[3]{1700} + \sqrt[3]{1700}}{40}$

Step 12.2.5.2

Add $2 \sqrt[3]{1700}$ and $\sqrt[3]{1700}$ .

$f (\frac{\sqrt[3]{1700}}{5}) = \frac{3 \sqrt[3]{1700}}{40}$

Step 12.2.6

The final answer is $\frac{3 \sqrt[3]{1700}}{40}$ .

$y = \frac{3 \sqrt[3]{1700}}{40}$

Step 13

These are the local extrema for $f (F) = 0.25 F + \frac{1.7}{F^{2}}$ .

$(\frac{\sqrt[3]{1700}}{5}, \frac{3 \sqrt[3]{1700}}{40})$ is a local minima

Step 14