Calculus Examples

$f (x) = 5 x^{\frac{7}{4}} - 70 x + 15$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $5 x^{\frac{7}{4}} - 70 x + 15$ with respect to $x$ is $\frac{d}{d x} [5 x^{\frac{7}{4}}] + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$ .

$\frac{d}{d x} [5 x^{\frac{7}{4}}] + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 1.2

Evaluate $\frac{d}{d x} [5 x^{\frac{7}{4}}]$ .

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Step 1.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{\frac{7}{4}}$ with respect to $x$ is $5 \frac{d}{d x} [x^{\frac{7}{4}}]$ .

$5 \frac{d}{d x} [x^{\frac{7}{4}}] + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{4}$ .

$5 (\frac{7}{4} x^{\frac{7}{4} - 1}) + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$5 (\frac{7}{4} x^{\frac{7}{4} - 1 \cdot \frac{4}{4}}) + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 1.2.4

Combine $- 1$ and $\frac{4}{4}$ .

$5 (\frac{7}{4} x^{\frac{7}{4} + \frac{- 1 \cdot 4}{4}}) + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 1.2.5

Combine the numerators over the common denominator.

$5 (\frac{7}{4} x^{\frac{7 - 1 \cdot 4}{4}}) + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $4$ .

$5 (\frac{7}{4} x^{\frac{7 - 4}{4}}) + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 1.2.6.2

Subtract $4$ from $7$ .

$5 (\frac{7}{4} x^{\frac{3}{4}}) + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 1.2.7

Combine $\frac{7}{4}$ and $x^{\frac{3}{4}}$ .

$5 \frac{7 x^{\frac{3}{4}}}{4} + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 1.2.8

Combine $5$ and $\frac{7 x^{\frac{3}{4}}}{4}$ .

$\frac{5 (7 x^{\frac{3}{4}})}{4} + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 1.2.9

Multiply $7$ by $5$ .

$\frac{35 x^{\frac{3}{4}}}{4} + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 1.3

Evaluate $\frac{d}{d x} [- 70 x]$ .

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Step 1.3.1

Since $- 70$ is constant with respect to $x$ , the derivative of $- 70 x$ with respect to $x$ is $- 70 \frac{d}{d x} [x]$ .

$\frac{35 x^{\frac{3}{4}}}{4} - 70 \frac{d}{d x} [x] + \frac{d}{d x} [15]$

Step 1.3.2

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

$\frac{35 x^{\frac{3}{4}}}{4} - 70 \cdot 1 + \frac{d}{d x} [15]$

Step 1.3.3

Multiply $- 70$ by $1$ .

$\frac{35 x^{\frac{3}{4}}}{4} - 70 + \frac{d}{d x} [15]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$\frac{35 x^{\frac{3}{4}}}{4} - 70 + 0$

Step 1.4.2

Add $\frac{35 x^{\frac{3}{4}}}{4} - 70$ and $0$ .

$\frac{35 x^{\frac{3}{4}}}{4} - 70$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $\frac{35 x^{\frac{3}{4}}}{4} - 70$ with respect to $x$ is $\frac{d}{d x} [\frac{35 x^{\frac{3}{4}}}{4}] + \frac{d}{d x} [- 70]$ .

$f'' (x) = \frac{d}{d x} (\frac{35 x^{\frac{3}{4}}}{4}) + \frac{d}{d x} (- 70)$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{35 x^{\frac{3}{4}}}{4}]$ .

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Step 2.2.1

Since $\frac{35}{4}$ is constant with respect to $x$ , the derivative of $\frac{35 x^{\frac{3}{4}}}{4}$ with respect to $x$ is $\frac{35}{4} \frac{d}{d x} [x^{\frac{3}{4}}]$ .

$f'' (x) = \frac{35}{4} \cdot \frac{d}{d x} (x^{\frac{3}{4}}) + \frac{d}{d x} (- 70)$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{4}$ .

$f'' (x) = \frac{35}{4} \cdot (\frac{3}{4} x^{\frac{3}{4} - 1}) + \frac{d}{d x} (- 70)$

Step 2.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f'' (x) = \frac{35}{4} \cdot (\frac{3}{4} x^{\frac{3}{4} - 1 \cdot \frac{4}{4}}) + \frac{d}{d x} (- 70)$

Step 2.2.4

Combine $- 1$ and $\frac{4}{4}$ .

$f'' (x) = \frac{35}{4} \cdot (\frac{3}{4} x^{\frac{3}{4} + \frac{- 1 \cdot 4}{4}}) + \frac{d}{d x} (- 70)$

Step 2.2.5

Combine the numerators over the common denominator.

$f'' (x) = \frac{35}{4} \cdot (\frac{3}{4} x^{\frac{3 - 1 \cdot 4}{4}}) + \frac{d}{d x} (- 70)$

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $- 1$ by $4$ .

$f'' (x) = \frac{35}{4} \cdot (\frac{3}{4} x^{\frac{3 - 4}{4}}) + \frac{d}{d x} (- 70)$

Step 2.2.6.2

Subtract $4$ from $3$ .

$f'' (x) = \frac{35}{4} \cdot (\frac{3}{4} x^{\frac{- 1}{4}}) + \frac{d}{d x} (- 70)$

Step 2.2.7

Move the negative in front of the fraction.

$f'' (x) = \frac{35}{4} \cdot (\frac{3}{4} x^{- \frac{1}{4}}) + \frac{d}{d x} (- 70)$

Step 2.2.8

Combine $\frac{3}{4}$ and $x^{- \frac{1}{4}}$ .

$f'' (x) = \frac{35}{4} \cdot \frac{3 x^{- \frac{1}{4}}}{4} + \frac{d}{d x} (- 70)$

Step 2.2.9

Multiply $\frac{35}{4}$ by $\frac{3 x^{- \frac{1}{4}}}{4}$ .

$f'' (x) = \frac{35 (3 x^{- \frac{1}{4}})}{4 \cdot 4} + \frac{d}{d x} (- 70)$

Step 2.2.10

Multiply $3$ by $35$ .

$f'' (x) = \frac{105 x^{- \frac{1}{4}}}{4 \cdot 4} + \frac{d}{d x} (- 70)$

Step 2.2.11

Multiply $4$ by $4$ .

$f'' (x) = \frac{105 x^{- \frac{1}{4}}}{16} + \frac{d}{d x} (- 70)$

Step 2.2.12

Move $x^{- \frac{1}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{105}{16 x^{\frac{1}{4}}} + \frac{d}{d x} (- 70)$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $- 70$ is constant with respect to $x$ , the derivative of $- 70$ with respect to $x$ is $0$ .

$f'' (x) = \frac{105}{16 x^{\frac{1}{4}}} + 0$

Step 2.3.2

Add $\frac{105}{16 x^{\frac{1}{4}}}$ and $0$ .

$f'' (x) = \frac{105}{16 x^{\frac{1}{4}}}$

Step 3

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

$\frac{35 x^{\frac{3}{4}}}{4} - 70 = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

By the Sum Rule, the derivative of $5 x^{\frac{7}{4}} - 70 x + 15$ with respect to $x$ is $\frac{d}{d x} [5 x^{\frac{7}{4}}] + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$ .

$\frac{d}{d x} [5 x^{\frac{7}{4}}] + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 4.1.2

Evaluate $\frac{d}{d x} [5 x^{\frac{7}{4}}]$ .

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Step 4.1.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{\frac{7}{4}}$ with respect to $x$ is $5 \frac{d}{d x} [x^{\frac{7}{4}}]$ .

$5 \frac{d}{d x} [x^{\frac{7}{4}}] + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 4.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{4}$ .

$5 (\frac{7}{4} x^{\frac{7}{4} - 1}) + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 4.1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$5 (\frac{7}{4} x^{\frac{7}{4} - 1 \cdot \frac{4}{4}}) + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 4.1.2.4

Combine $- 1$ and $\frac{4}{4}$ .

$5 (\frac{7}{4} x^{\frac{7}{4} + \frac{- 1 \cdot 4}{4}}) + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 4.1.2.5

Combine the numerators over the common denominator.

$5 (\frac{7}{4} x^{\frac{7 - 1 \cdot 4}{4}}) + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 4.1.2.6

Simplify the numerator.

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Step 4.1.2.6.1

Multiply $- 1$ by $4$ .

$5 (\frac{7}{4} x^{\frac{7 - 4}{4}}) + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 4.1.2.6.2

Subtract $4$ from $7$ .

$5 (\frac{7}{4} x^{\frac{3}{4}}) + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 4.1.2.7

Combine $\frac{7}{4}$ and $x^{\frac{3}{4}}$ .

$5 \frac{7 x^{\frac{3}{4}}}{4} + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 4.1.2.8

Combine $5$ and $\frac{7 x^{\frac{3}{4}}}{4}$ .

$\frac{5 (7 x^{\frac{3}{4}})}{4} + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 4.1.2.9

Multiply $7$ by $5$ .

$\frac{35 x^{\frac{3}{4}}}{4} + \frac{d}{d x} [- 70 x] + \frac{d}{d x} [15]$

Step 4.1.3

Evaluate $\frac{d}{d x} [- 70 x]$ .

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Step 4.1.3.1

Since $- 70$ is constant with respect to $x$ , the derivative of $- 70 x$ with respect to $x$ is $- 70 \frac{d}{d x} [x]$ .

$\frac{35 x^{\frac{3}{4}}}{4} - 70 \frac{d}{d x} [x] + \frac{d}{d x} [15]$

Step 4.1.3.2

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

$\frac{35 x^{\frac{3}{4}}}{4} - 70 \cdot 1 + \frac{d}{d x} [15]$

Step 4.1.3.3

Multiply $- 70$ by $1$ .

$\frac{35 x^{\frac{3}{4}}}{4} - 70 + \frac{d}{d x} [15]$

Step 4.1.4

Differentiate using the Constant Rule.

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Step 4.1.4.1

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$\frac{35 x^{\frac{3}{4}}}{4} - 70 + 0$

Step 4.1.4.2

Add $\frac{35 x^{\frac{3}{4}}}{4} - 70$ and $0$ .

$f' (x) = \frac{35 x^{\frac{3}{4}}}{4} - 70$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{35 x^{\frac{3}{4}}}{4} - 70$ .

$\frac{35 x^{\frac{3}{4}}}{4} - 70$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{35 x^{\frac{3}{4}}}{4} - 70 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{35 x^{\frac{3}{4}}}{4} - 70 = 0$

Step 5.2

Add $70$ to both sides of the equation.

$\frac{35 x^{\frac{3}{4}}}{4} = 70$

Step 5.3

Multiply both sides of the equation by $\frac{4}{35}$ .

$\frac{4}{35} \cdot \frac{35 x^{\frac{3}{4}}}{4} = \frac{4}{35} \cdot 70$

Step 5.4

Simplify both sides of the equation.

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Step 5.4.1

Simplify the left side.

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Step 5.4.1.1

Simplify $\frac{4}{35} \cdot \frac{35 x^{\frac{3}{4}}}{4}$ .

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Step 5.4.1.1.1

Combine.

$\frac{4 (35 x^{\frac{3}{4}})}{35 \cdot 4} = \frac{4}{35} \cdot 70$

Step 5.4.1.1.2

Cancel the common factor.

$\frac{4 (35 x^{\frac{3}{4}})}{35 \cdot 4} = \frac{4}{35} \cdot 70$

Step 5.4.1.1.3

Rewrite the expression.

$\frac{35 x^{\frac{3}{4}}}{35} = \frac{4}{35} \cdot 70$

Step 5.4.1.1.4

Cancel the common factor.

$\frac{35 x^{\frac{3}{4}}}{35} = \frac{4}{35} \cdot 70$

Step 5.4.1.1.5

Divide $x^{\frac{3}{4}}$ by $1$ .

$x^{\frac{3}{4}} = \frac{4}{35} \cdot 70$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify $\frac{4}{35} \cdot 70$ .

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Step 5.4.2.1.1

Cancel the common factor of $35$ .

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Step 5.4.2.1.1.1

Factor $35$ out of $70$ .

$x^{\frac{3}{4}} = \frac{4}{35} \cdot (35 (2))$

Step 5.4.2.1.1.2

Cancel the common factor.

$x^{\frac{3}{4}} = \frac{4}{35} \cdot (35 \cdot 2)$

Step 5.4.2.1.1.3

Rewrite the expression.

$x^{\frac{3}{4}} = 4 \cdot 2$

Step 5.4.2.1.2

Multiply $4$ by $2$ .

$x^{\frac{3}{4}} = 8$

Step 5.5

Raise each side of the equation to the power of $\frac{4}{3}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{3}{4}})}^{\frac{4}{3}} = 8^{\frac{4}{3}}$

Step 5.6

Simplify the exponent.

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Step 5.6.1

Simplify the left side.

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Step 5.6.1.1

Simplify ${(x^{\frac{3}{4}})}^{\frac{4}{3}}$ .

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Step 5.6.1.1.1

Multiply the exponents in ${(x^{\frac{3}{4}})}^{\frac{4}{3}}$ .

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Step 5.6.1.1.1.1

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

$x^{\frac{3}{4} \cdot \frac{4}{3}} = 8^{\frac{4}{3}}$

Step 5.6.1.1.1.2

Cancel the common factor of $3$ .

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Step 5.6.1.1.1.2.1

Cancel the common factor.

$x^{\frac{3}{4} \cdot \frac{4}{3}} = 8^{\frac{4}{3}}$

Step 5.6.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{4} \cdot 4} = 8^{\frac{4}{3}}$

Step 5.6.1.1.1.3

Cancel the common factor of $4$ .

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Step 5.6.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = 8^{\frac{4}{3}}$

Step 5.6.1.1.1.3.2

Rewrite the expression.

$x^{1} = 8^{\frac{4}{3}}$

Step 5.6.1.1.2

Simplify.

$x = 8^{\frac{4}{3}}$

Step 5.6.2

Simplify the right side.

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Step 5.6.2.1

Simplify $8^{\frac{4}{3}}$ .

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Step 5.6.2.1.1

Simplify the expression.

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Step 5.6.2.1.1.1

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

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

Step 5.6.2.1.1.2

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

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

Step 5.6.2.1.2

Cancel the common factor of $3$ .

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Step 5.6.2.1.2.1

Cancel the common factor.

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

Step 5.6.2.1.2.2

Rewrite the expression.

$x = 2^{4}$

Step 5.6.2.1.3

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

$x = 16$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

$\frac{35 \sqrt[4]{x^{3}}}{4} - 70$

Step 6.2

Set the radicand in $\sqrt[4]{x^{3}}$ less than $0$ to find where the expression is undefined.

$x^{3} < 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

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

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

Step 6.3.2

Simplify the equation.

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Step 6.3.2.1

Simplify the left side.

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Step 6.3.2.1.1

Pull terms out from under the radical.

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

Step 6.3.2.2

Simplify the right side.

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Step 6.3.2.2.1

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

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Step 6.3.2.2.1.1

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

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

Step 6.3.2.2.1.2

Pull terms out from under the radical.

$x < 0$

Step 6.4

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

$x < 0$

$(- \infty, 0)$

$x < 0$

$(- \infty, 0)$

Step 7

Critical points to evaluate.

$x = 16$

Step 8

Evaluate the second derivative at $x = 16$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{105}{16 {(16)}^{\frac{1}{4}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Multiply $16$ by ${(16)}^{\frac{1}{4}}$ by adding the exponents.

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Step 9.1.1

Multiply $16$ by ${(16)}^{\frac{1}{4}}$ .

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Step 9.1.1.1

Raise $16$ to the power of $1$ .

$\frac{105}{16^{1} {(16)}^{\frac{1}{4}}}$

Step 9.1.1.2

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

$\frac{105}{16^{1 + \frac{1}{4}}}$

Step 9.1.2

Write $1$ as a fraction with a common denominator.

$\frac{105}{16^{\frac{4}{4} + \frac{1}{4}}}$

Step 9.1.3

Combine the numerators over the common denominator.

$\frac{105}{16^{\frac{4 + 1}{4}}}$

Step 9.1.4

Add $4$ and $1$ .

$\frac{105}{16^{\frac{5}{4}}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Rewrite $16$ as $2^{4}$ .

$\frac{105}{{(2^{4})}^{\frac{5}{4}}}$

Step 9.2.2

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

$\frac{105}{2^{4 (\frac{5}{4})}}$

Step 9.2.3

Cancel the common factor of $4$ .

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Step 9.2.3.1

Cancel the common factor.

$\frac{105}{2^{4 (\frac{5}{4})}}$

Step 9.2.3.2

Rewrite the expression.

$\frac{105}{2^{5}}$

Step 9.2.4

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

$\frac{105}{32}$

Step 10

$x = 16$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 16$ is a local minimum

Step 11

Find the y-value when $x = 16$ .

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Step 11.1

Replace the variable $x$ with $16$ in the expression.

$f (16) = 5 {(16)}^{\frac{7}{4}} - 70 \cdot 16 + 15$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Rewrite $16$ as $2^{4}$ .

$f (16) = 5 {(2^{4})}^{\frac{7}{4}} - 70 \cdot 16 + 15$

Step 11.2.1.2

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

$f (16) = 5 \cdot 2^{4 (\frac{7}{4})} - 70 \cdot 16 + 15$

Step 11.2.1.3

Cancel the common factor of $4$ .

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Step 11.2.1.3.1

Cancel the common factor.

$f (16) = 5 \cdot 2^{4 (\frac{7}{4})} - 70 \cdot 16 + 15$

Step 11.2.1.3.2

Rewrite the expression.

$f (16) = 5 \cdot 2^{7} - 70 \cdot 16 + 15$

Step 11.2.1.4

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

$f (16) = 5 \cdot 128 - 70 \cdot 16 + 15$

Step 11.2.1.5

Multiply $5$ by $128$ .

$f (16) = 640 - 70 \cdot 16 + 15$

Step 11.2.1.6

Multiply $- 70$ by $16$ .

$f (16) = 640 - 1120 + 15$

Step 11.2.2

Simplify by adding and subtracting.

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Step 11.2.2.1

Subtract $1120$ from $640$ .

$f (16) = - 480 + 15$

Step 11.2.2.2

Add $- 480$ and $15$ .

$f (16) = - 465$

Step 11.2.3

The final answer is $- 465$ .

$y = - 465$

Step 12

These are the local extrema for $f (x) = 5 x^{\frac{7}{4}} - 70 x + 15$ .

$(16, - 465)$ is a local minima

Step 13