Calculus Examples

$f (x) = x + \frac{16}{x}$ , $[4, 20]$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

Differentiate.

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Step 1.1.1.1.1

By the Sum Rule, the derivative of $x + \frac{16}{x}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{16}{x}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [\frac{16}{x}]$

Step 1.1.1.1.2

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

$1 + \frac{d}{d x} [\frac{16}{x}]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [\frac{16}{x}]$ .

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Step 1.1.1.2.1

Since $16$ is constant with respect to $x$ , the derivative of $\frac{16}{x}$ with respect to $x$ is $16 \frac{d}{d x} [\frac{1}{x}]$ .

$1 + 16 \frac{d}{d x} [\frac{1}{x}]$

Step 1.1.1.2.2

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

$1 + 16 \frac{d}{d x} [x^{- 1}]$

Step 1.1.1.2.3

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

$1 + 16 (- x^{- 2})$

Step 1.1.1.2.4

Multiply $- 1$ by $16$ .

$1 - 16 x^{- 2}$

Step 1.1.1.3

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

$1 - 16 \frac{1}{x^{2}}$

Step 1.1.1.4

Simplify.

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Step 1.1.1.4.1

Combine terms.

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Step 1.1.1.4.1.1

Combine $- 16$ and $\frac{1}{x^{2}}$ .

$1 + \frac{- 16}{x^{2}}$

Step 1.1.1.4.1.2

Move the negative in front of the fraction.

$1 - \frac{16}{x^{2}}$

Step 1.1.1.4.2

Reorder terms.

$f' (x) = - \frac{16}{x^{2}} + 1$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{16}{x^{2}} + 1$ .

$- \frac{16}{x^{2}} + 1$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $- \frac{16}{x^{2}} + 1 = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$- \frac{16}{x^{2}} + 1 = 0$

Step 1.2.2

Subtract $1$ from both sides of the equation.

$- \frac{16}{x^{2}} = - 1$

Step 1.2.3

Find the LCD of the terms in the equation.

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Step 1.2.3.1

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

$x^{2}, 1$

Step 1.2.3.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 1.2.4

Multiply each term in $- \frac{16}{x^{2}} = - 1$ by $x^{2}$ to eliminate the fractions.

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Step 1.2.4.1

Multiply each term in $- \frac{16}{x^{2}} = - 1$ by $x^{2}$ .

$- \frac{16}{x^{2}} x^{2} = - x^{2}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

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

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Step 1.2.4.2.1.1

Move the leading negative in $- \frac{16}{x^{2}}$ into the numerator.

$\frac{- 16}{x^{2}} x^{2} = - x^{2}$

Step 1.2.4.2.1.2

Cancel the common factor.

$\frac{- 16}{x^{2}} x^{2} = - x^{2}$

Step 1.2.4.2.1.3

Rewrite the expression.

$- 16 = - x^{2}$

Step 1.2.5

Solve the equation.

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Step 1.2.5.1

Rewrite the equation as $- x^{2} = - 16$ .

$- x^{2} = - 16$

Step 1.2.5.2

Divide each term in $- x^{2} = - 16$ by $- 1$ and simplify.

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Step 1.2.5.2.1

Divide each term in $- x^{2} = - 16$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 16}{- 1}$

Step 1.2.5.2.2

Simplify the left side.

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Step 1.2.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 16}{- 1}$

Step 1.2.5.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 16}{- 1}$

Step 1.2.5.2.3

Simplify the right side.

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Step 1.2.5.2.3.1

Divide $- 16$ by $- 1$ .

$x^{2} = 16$

Step 1.2.5.3

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

$x = \pm \sqrt{16}$

Step 1.2.5.4

Simplify $\pm \sqrt{16}$ .

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Step 1.2.5.4.1

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

$x = \pm \sqrt{4^{2}}$

Step 1.2.5.4.2

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

$x = \pm 4$

Step 1.2.5.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.5.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 4$

Step 1.2.5.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 4$

Step 1.2.5.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 4, - 4$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Set the denominator in $\frac{16}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 1.3.2

Solve for $x$ .

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Step 1.3.2.1

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

$x = \pm \sqrt{0}$

Step 1.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.3.2.2.1

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

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

Step 1.3.2.2.2

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

$x = \pm 0$

Step 1.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.4

Evaluate $x + \frac{16}{x}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 4$ .

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Step 1.4.1.1

Substitute $4$ for $x$ .

$(4) + \frac{16}{4}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Divide $16$ by $4$ .

$4 + 4$

Step 1.4.1.2.2

Add $4$ and $4$ .

$8$

Step 1.4.2

Evaluate at $x = - 4$ .

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Step 1.4.2.1

Substitute $- 4$ for $x$ .

$(- 4) + \frac{16}{- 4}$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Divide $16$ by $- 4$ .

$- 4 - 4$

Step 1.4.2.2.2

Subtract $4$ from $- 4$ .

$- 8$

Step 1.4.3

Evaluate at $x = 0$ .

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Step 1.4.3.1

Substitute $0$ for $x$ .

$(0) + \frac{16}{0}$

Step 1.4.3.2

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

Undefined

Step 1.4.4

List all of the points.

$(4, 8), (- 4, - 8)$

Step 2

Exclude the points that are not on the interval.

$(4, 8)$

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = 4$ .

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Step 3.1.1

Substitute $4$ for $x$ .

$(4) + \frac{16}{4}$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Divide $16$ by $4$ .

$4 + 4$

Step 3.1.2.2

Add $4$ and $4$ .

$8$

Step 3.2

Evaluate at $x = 20$ .

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Step 3.2.1

Substitute $20$ for $x$ .

$(20) + \frac{16}{20}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Cancel the common factor of $16$ and $20$ .

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Step 3.2.2.1.1

Factor $4$ out of $16$ .

$20 + \frac{4 (4)}{20}$

Step 3.2.2.1.2

Cancel the common factors.

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Step 3.2.2.1.2.1

Factor $4$ out of $20$ .

$20 + \frac{4 \cdot 4}{4 \cdot 5}$

Step 3.2.2.1.2.2

Cancel the common factor.

$20 + \frac{4 \cdot 4}{4 \cdot 5}$

Step 3.2.2.1.2.3

Rewrite the expression.

$20 + \frac{4}{5}$

Step 3.2.2.2

To write $20$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$20 \cdot \frac{5}{5} + \frac{4}{5}$

Step 3.2.2.3

Combine $20$ and $\frac{5}{5}$ .

$\frac{20 \cdot 5}{5} + \frac{4}{5}$

Step 3.2.2.4

Combine the numerators over the common denominator.

$\frac{20 \cdot 5 + 4}{5}$

Step 3.2.2.5

Simplify the numerator.

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Step 3.2.2.5.1

Multiply $20$ by $5$ .

$\frac{100 + 4}{5}$

Step 3.2.2.5.2

Add $100$ and $4$ .

$\frac{104}{5}$

Step 3.3

List all of the points.

$(4, 8), (20, \frac{104}{5})$

Step 4

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(20, \frac{104}{5})$

Absolute Minimum: $(4, 8)$

Step 5