Calculus Examples

$f (x) = 8.4 x^{0.75} - 2.1 x + 38.75$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $8.4 x^{0.75} - 2.1 x + 38.75$ with respect to $x$ is $\frac{d}{d x} [8.4 x^{0.75}] + \frac{d}{d x} [- 2.1 x] + \frac{d}{d x} [38.75]$ .

$\frac{d}{d x} [8.4 x^{0.75}] + \frac{d}{d x} [- 2.1 x] + \frac{d}{d x} [38.75]$

Step 1.2

Evaluate $\frac{d}{d x} [8.4 x^{0.75}]$ .

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Step 1.2.1

Since $8.4$ is constant with respect to $x$ , the derivative of $8.4 x^{0.75}$ with respect to $x$ is $8.4 \frac{d}{d x} [x^{0.75}]$ .

$8.4 \frac{d}{d x} [x^{0.75}] + \frac{d}{d x} [- 2.1 x] + \frac{d}{d x} [38.75]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 0.75$ .

$8.4 (0.75 x^{- 0.25}) + \frac{d}{d x} [- 2.1 x] + \frac{d}{d x} [38.75]$

Step 1.2.3

Multiply $0.75$ by $8.4$ .

$6.3 x^{- 0.25} + \frac{d}{d x} [- 2.1 x] + \frac{d}{d x} [38.75]$

Step 1.3

Evaluate $\frac{d}{d x} [- 2.1 x]$ .

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Step 1.3.1

Since $- 2.1$ is constant with respect to $x$ , the derivative of $- 2.1 x$ with respect to $x$ is $- 2.1 \frac{d}{d x} [x]$ .

$6.3 x^{- 0.25} - 2.1 \frac{d}{d x} [x] + \frac{d}{d x} [38.75]$

Step 1.3.2

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

$6.3 x^{- 0.25} - 2.1 \cdot 1 + \frac{d}{d x} [38.75]$

Step 1.3.3

Multiply $- 2.1$ by $1$ .

$6.3 x^{- 0.25} - 2.1 + \frac{d}{d x} [38.75]$

Step 1.4

Since $38.75$ is constant with respect to $x$ , the derivative of $38.75$ with respect to $x$ is $0$ .

$6.3 x^{- 0.25} - 2.1 + 0$

Step 1.5

Simplify.

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Step 1.5.1

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

$6.3 \frac{1}{x^{0.25}} - 2.1 + 0$

Step 1.5.2

Combine terms.

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Step 1.5.2.1

Combine $6.3$ and $\frac{1}{x^{0.25}}$ .

$\frac{6.3}{x^{0.25}} - 2.1 + 0$

Step 1.5.2.2

Add $\frac{6.3}{x^{0.25}} - 2.1$ and $0$ .

$\frac{6.3}{x^{0.25}} - 2.1$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $\frac{6.3}{x^{0.25}} - 2.1$ with respect to $x$ is $\frac{d}{d x} [\frac{6.3}{x^{0.25}}] + \frac{d}{d x} [- 2.1]$ .

$f'' (x) = \frac{d}{d x} (\frac{6.3}{x^{0.25}}) + \frac{d}{d x} (- 2.1)$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{6.3}{x^{0.25}}]$ .

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Step 2.2.1

Since $6.3$ is constant with respect to $x$ , the derivative of $\frac{6.3}{x^{0.25}}$ with respect to $x$ is $6.3 \frac{d}{d x} [\frac{1}{x^{0.25}}]$ .

$f'' (x) = 6.3 \frac{d}{d x} (\frac{1}{x^{0.25}}) + \frac{d}{d x} (- 2.1)$

Step 2.2.2

Rewrite $\frac{1}{x^{0.25}}$ as ${(x^{0.25})}^{- 1}$ .

$f'' (x) = 6.3 \frac{d}{d x} ({(x^{0.25})}^{- 1}) + \frac{d}{d x} (- 2.1)$

Step 2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{0.25}$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u$ as $x^{0.25}$ .

$f'' (x) = 6.3 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{0.25})) + \frac{d}{d x} (- 2.1)$

Step 2.2.3.2

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

$f'' (x) = 6.3 (- u^{- 2} \frac{d}{d x} x^{0.25}) + \frac{d}{d x} (- 2.1)$

Step 2.2.3.3

Replace all occurrences of $u$ with $x^{0.25}$ .

$f'' (x) = 6.3 (- {(x^{0.25})}^{- 2} \frac{d}{d x} x^{0.25}) + \frac{d}{d x} (- 2.1)$

Step 2.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 0.25$ .

$f'' (x) = 6.3 (- {(x^{0.25})}^{- 2} (0.25 x^{- 0.75})) + \frac{d}{d x} (- 2.1)$

Step 2.2.5

Multiply the exponents in ${(x^{0.25})}^{- 2}$ .

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Step 2.2.5.1

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

$f'' (x) = 6.3 (- x^{0.25 \cdot - 2} (0.25 x^{- 0.75})) + \frac{d}{d x} (- 2.1)$

Step 2.2.5.2

Multiply $0.25$ by $- 2$ .

$f'' (x) = 6.3 (- x^{- 0.5} (0.25 x^{- 0.75})) + \frac{d}{d x} (- 2.1)$

Step 2.2.6

Multiply $0.25$ by $- 1$ .

$f'' (x) = 6.3 (- 0.25 x^{- 0.5} x^{- 0.75}) + \frac{d}{d x} (- 2.1)$

Step 2.2.7

Multiply $x^{- 0.5}$ by $x^{- 0.75}$ by adding the exponents.

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Step 2.2.7.1

Move $x^{- 0.75}$ .

$f'' (x) = 6.3 (- 0.25 (x^{- 0.75} x^{- 0.5})) + \frac{d}{d x} (- 2.1)$

Step 2.2.7.2

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

$f'' (x) = 6.3 (- 0.25 x^{- 0.75 - 0.5}) + \frac{d}{d x} (- 2.1)$

Step 2.2.7.3

Subtract $0.5$ from $- 0.75$ .

$f'' (x) = 6.3 (- 0.25 x^{- 1.25}) + \frac{d}{d x} (- 2.1)$

Step 2.2.8

Multiply $- 0.25$ by $6.3$ .

$f'' (x) = - 1.575 x^{- 1.25} + \frac{d}{d x} (- 2.1)$

Step 2.3

Since $- 2.1$ is constant with respect to $x$ , the derivative of $- 2.1$ with respect to $x$ is $0$ .

$f'' (x) = - 1.575 x^{- 1.25} + 0$

Step 2.4

Simplify.

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Step 2.4.1

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

$f'' (x) = - 1.575 \frac{1}{x^{1.25}} + 0$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Combine $- 1.575$ and $\frac{1}{x^{1.25}}$ .

$f'' (x) = \frac{- 1.575}{x^{1.25}} + 0$

Step 2.4.2.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{1.575}{x^{1.25}} + 0$

Step 2.4.2.3

Add $- \frac{1.575}{x^{1.25}}$ and $0$ .

$f'' (x) = - \frac{1.575}{x^{1.25}}$

Step 3

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

$\frac{6.3}{x^{0.25}} - 2.1 = 0$

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 $8.4 x^{0.75} - 2.1 x + 38.75$ with respect to $x$ is $\frac{d}{d x} [8.4 x^{0.75}] + \frac{d}{d x} [- 2.1 x] + \frac{d}{d x} [38.75]$ .

$\frac{d}{d x} [8.4 x^{0.75}] + \frac{d}{d x} [- 2.1 x] + \frac{d}{d x} [38.75]$

Step 4.1.2

Evaluate $\frac{d}{d x} [8.4 x^{0.75}]$ .

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Step 4.1.2.1

Since $8.4$ is constant with respect to $x$ , the derivative of $8.4 x^{0.75}$ with respect to $x$ is $8.4 \frac{d}{d x} [x^{0.75}]$ .

$8.4 \frac{d}{d x} [x^{0.75}] + \frac{d}{d x} [- 2.1 x] + \frac{d}{d x} [38.75]$

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 $n = 0.75$ .

$8.4 (0.75 x^{- 0.25}) + \frac{d}{d x} [- 2.1 x] + \frac{d}{d x} [38.75]$

Step 4.1.2.3

Multiply $0.75$ by $8.4$ .

$6.3 x^{- 0.25} + \frac{d}{d x} [- 2.1 x] + \frac{d}{d x} [38.75]$

Step 4.1.3

Evaluate $\frac{d}{d x} [- 2.1 x]$ .

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Step 4.1.3.1

Since $- 2.1$ is constant with respect to $x$ , the derivative of $- 2.1 x$ with respect to $x$ is $- 2.1 \frac{d}{d x} [x]$ .

$6.3 x^{- 0.25} - 2.1 \frac{d}{d x} [x] + \frac{d}{d x} [38.75]$

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 $n = 1$ .

$6.3 x^{- 0.25} - 2.1 \cdot 1 + \frac{d}{d x} [38.75]$

Step 4.1.3.3

Multiply $- 2.1$ by $1$ .

$6.3 x^{- 0.25} - 2.1 + \frac{d}{d x} [38.75]$

Step 4.1.4

Since $38.75$ is constant with respect to $x$ , the derivative of $38.75$ with respect to $x$ is $0$ .

$6.3 x^{- 0.25} - 2.1 + 0$

Step 4.1.5

Simplify.

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Step 4.1.5.1

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

$6.3 \frac{1}{x^{0.25}} - 2.1 + 0$

Step 4.1.5.2

Combine terms.

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Step 4.1.5.2.1

Combine $6.3$ and $\frac{1}{x^{0.25}}$ .

$\frac{6.3}{x^{0.25}} - 2.1 + 0$

Step 4.1.5.2.2

Add $\frac{6.3}{x^{0.25}} - 2.1$ and $0$ .

$f' (x) = \frac{6.3}{x^{0.25}} - 2.1$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{6.3}{x^{0.25}} - 2.1$ .

$\frac{6.3}{x^{0.25}} - 2.1$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{6.3}{x^{0.25}} - 2.1 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{6.3}{x^{0.25}} - 2.1 = 0$

Step 5.2

Add $2.1$ to both sides of the equation.

$\frac{6.3}{x^{0.25}} = 2.1$

Step 5.3

Find the LCD of the terms in the equation.

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Step 5.3.1

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

$x^{0.25}, 1$

Step 5.3.2

The LCM of one and any expression is the expression.

$x^{0.25}$

Step 5.4

Multiply each term in $\frac{6.3}{x^{0.25}} = 2.1$ by $x^{0.25}$ to eliminate the fractions.

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Step 5.4.1

Multiply each term in $\frac{6.3}{x^{0.25}} = 2.1$ by $x^{0.25}$ .

$\frac{6.3}{x^{0.25}} x^{0.25} = 2.1 x^{0.25}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $x^{0.25}$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{6.3}{x^{0.25}} x^{0.25} = 2.1 x^{0.25}$

Step 5.4.2.1.2

Rewrite the expression.

$6.3 = 2.1 x^{0.25}$

Step 5.5

Solve the equation.

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Step 5.5.1

Rewrite the equation as $2.1 x^{0.25} = 6.3$ .

$2.1 x^{0.25} = 6.3$

Step 5.5.2

Divide each term in $2.1 x^{0.25} = 6.3$ by $2.1$ and simplify.

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Step 5.5.2.1

Divide each term in $2.1 x^{0.25} = 6.3$ by $2.1$ .

$\frac{2.1 x^{0.25}}{2.1} = \frac{6.3}{2.1}$

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

Cancel the common factor of $2.1$ .

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Step 5.5.2.2.1.1

Cancel the common factor.

$\frac{2.1 x^{0.25}}{2.1} = \frac{6.3}{2.1}$

Step 5.5.2.2.1.2

Divide $x^{0.25}$ by $1$ .

$x^{0.25} = \frac{6.3}{2.1}$

Step 5.5.2.3

Simplify the right side.

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Step 5.5.2.3.1

Divide $6.3$ by $2.1$ .

$x^{0.25} = 3$

Step 5.5.3

Convert the decimal exponent to a fractional exponent.

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Step 5.5.3.1

Convert the decimal number to a fraction by placing the decimal number over a power of ten. Since there are $2$ numbers to the right of the decimal point, place the decimal number over $10^{2}$ $(100)$ . Next, add the whole number to the left of the decimal.

$x^{0 \frac{25}{100}} = 3$

Step 5.5.3.2

Reduce the fraction.

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Step 5.5.3.2.1

Convert $0 \frac{25}{100}$ to an improper fraction.

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Step 5.5.3.2.1.1

A mixed number is an addition of its whole and fractional parts.

$x^{0 + \frac{25}{100}} = 3$

Step 5.5.3.2.1.2

Add $0$ and $\frac{25}{100}$ .

$x^{\frac{25}{100}} = 3$

Step 5.5.3.2.2

Cancel the common factor of $25$ and $100$ .

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Step 5.5.3.2.2.1

Factor $25$ out of $25$ .

$x^{\frac{25 (1)}{100}} = 3$

Step 5.5.3.2.2.2

Cancel the common factors.

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Step 5.5.3.2.2.2.1

Factor $25$ out of $100$ .

$x^{\frac{25 \cdot 1}{25 \cdot 4}} = 3$

Step 5.5.3.2.2.2.2

Cancel the common factor.

$x^{\frac{25 \cdot 1}{25 \cdot 4}} = 3$

Step 5.5.3.2.2.2.3

Rewrite the expression.

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

Step 5.5.4

Raise each side of the equation to the power of $\frac{1}{0.25}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{4}})}^{\frac{1}{0.25}} = 3^{\frac{1}{0.25}}$

Step 5.5.5

Simplify the exponent.

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Step 5.5.5.1

Simplify the left side.

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Step 5.5.5.1.1

Simplify ${(x^{\frac{1}{4}})}^{\frac{1}{0.25}}$ .

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Step 5.5.5.1.1.1

Multiply the exponents in ${(x^{\frac{1}{4}})}^{\frac{1}{0.25}}$ .

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Step 5.5.5.1.1.1.1

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

$x^{\frac{1}{4} \cdot \frac{1}{0.25}} = 3^{\frac{1}{0.25}}$

Step 5.5.5.1.1.1.2

Cancel the common factor of $0.25$ .

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Step 5.5.5.1.1.1.2.1

Factor $0.25$ out of $1$ .

$x^{\frac{0.25 (4)}{4} \cdot \frac{1}{0.25}} = 3^{\frac{1}{0.25}}$

Step 5.5.5.1.1.1.2.2

Cancel the common factor.

$x^{\frac{0.25 \cdot 4}{4} \cdot \frac{1}{0.25}} = 3^{\frac{1}{0.25}}$

Step 5.5.5.1.1.1.2.3

Rewrite the expression.

$x^{\frac{4}{4}} = 3^{\frac{1}{0.25}}$

Step 5.5.5.1.1.1.3

Divide $4$ by $4$ .

$x^{1} = 3^{\frac{1}{0.25}}$

Step 5.5.5.1.1.2

Simplify.

$x = 3^{\frac{1}{0.25}}$

Step 5.5.5.2

Simplify the right side.

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Step 5.5.5.2.1

Simplify $3^{\frac{1}{0.25}}$ .

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Step 5.5.5.2.1.1

Divide $1$ by $0.25$ .

$x = 3^{4}$

Step 5.5.5.2.1.2

Raise $3$ to the power of $4$ .

$x = 81$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

Change $0.25$ into a fraction.

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Step 6.1.1.1

Multiply by $\frac{100}{100}$ to remove the decimal.

$\frac{6.3}{x^{\frac{100 \cdot 0.25}{100}}} - 2.1$

Step 6.1.1.2

Multiply $100$ by $0.25$ .

$\frac{6.3}{x^{\frac{25}{100}}} - 2.1$

Step 6.1.1.3

Cancel the common factor of $25$ and $100$ .

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Step 6.1.1.3.1

Factor $25$ out of $25$ .

$\frac{6.3}{x^{\frac{25 (1)}{100}}} - 2.1$

Step 6.1.1.3.2

Cancel the common factors.

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Step 6.1.1.3.2.1

Factor $25$ out of $100$ .

$\frac{6.3}{x^{\frac{25 \cdot 1}{25 \cdot 4}}} - 2.1$

Step 6.1.1.3.2.2

Cancel the common factor.

$\frac{6.3}{x^{\frac{25 \cdot 1}{25 \cdot 4}}} - 2.1$

Step 6.1.1.3.2.3

Rewrite the expression.

$\frac{6.3}{x^{\frac{1}{4}}} - 2.1$

Step 6.1.2

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

$\frac{6.3}{\sqrt[4]{x^{1}}} - 2.1$

Step 6.1.3

Anything raised to $1$ is the base itself.

$\frac{6.3}{\sqrt[4]{x}} - 2.1$

Step 6.2

Set the denominator in $\frac{6.3}{\sqrt[4]{x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[4]{x} = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{x}}^{4} = 0^{4}$

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

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

${(x^{\frac{1}{4}})}^{4} = 0^{4}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Simplify ${(x^{\frac{1}{4}})}^{4}$ .

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Step 6.3.2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{4}})}^{4}$ .

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Step 6.3.2.2.1.1.1

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

$x^{\frac{1}{4} \cdot 4} = 0^{4}$

Step 6.3.2.2.1.1.2

Cancel the common factor of $4$ .

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Step 6.3.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = 0^{4}$

Step 6.3.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{4}$

Step 6.3.2.2.1.2

Simplify.

$x = 0^{4}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$x = 0$

Step 6.4

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

$x < 0$

Step 6.5

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 $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 7

Critical points to evaluate.

$x = 81, 0$

Step 8

Evaluate the second derivative at $x = 81$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{1.575}{{(81)}^{1.25}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Raise $81$ to the power of $1.25$ .

$- \frac{1.575}{243}$

Step 9.2

Divide $1.575$ by $243$ .

$- 1 \cdot 0.006\overline{481}$

Step 9.3

Multiply $- 1$ by $0.006\overline{481}$ .

$- 0.006\overline{481}$

Step 10

$x = 81$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 81$ is a local maximum

Step 11

Find the y-value when $x = 81$ .

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Step 11.1

Replace the variable $x$ with $81$ in the expression.

$f (81) = 8.4 {(81)}^{0.75} - 2.1 \cdot 81 + 38.75$

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

Raise $81$ to the power of $0.75$ .

$f (81) = 8.4 \cdot 27 - 2.1 \cdot 81 + 38.75$

Step 11.2.1.2

Multiply $8.4$ by $27$ .

$f (81) = 226.8 - 2.1 \cdot 81 + 38.75$

Step 11.2.1.3

Multiply $- 2.1$ by $81$ .

$f (81) = 226.8 - 170.1 + 38.75$

Step 11.2.2

Simplify by adding and subtracting.

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Step 11.2.2.1

Subtract $170.1$ from $226.8$ .

$f (81) = 56.7 + 38.75$

Step 11.2.2.2

Add $56.7$ and $38.75$ .

$f (81) = 95.45$

Step 11.2.3

The final answer is $95.45$ .

$y = 95.45$

Step 12

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{1.575}{{(0)}^{1.25}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Raising $0$ to any positive power yields $0$ .

$- \frac{1.575}{0}$

Step 13.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 14

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 15