Calculus Examples

$f (x) = 2 x - \frac{5}{x}$ on $5$ , $7$

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

By the Sum Rule, the derivative of $2 x - \frac{5}{x}$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{5}{x}]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{5}{x}]$

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{5}{x}]$

Step 1.1.1.2.2

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

$2 \cdot 1 + \frac{d}{d x} [- \frac{5}{x}]$

Step 1.1.1.2.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- \frac{5}{x}]$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [- \frac{5}{x}]$ .

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Step 1.1.1.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- \frac{5}{x}$ with respect to $x$ is $- 5 \frac{d}{d x} [\frac{1}{x}]$ .

$2 - 5 \frac{d}{d x} [\frac{1}{x}]$

Step 1.1.1.3.2

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

$2 - 5 \frac{d}{d x} [x^{- 1}]$

Step 1.1.1.3.3

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

$2 - 5 (- x^{- 2})$

Step 1.1.1.3.4

Multiply $- 1$ by $- 5$ .

$2 + 5 x^{- 2}$

Step 1.1.1.4

Simplify.

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Step 1.1.1.4.1

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

$2 + 5 \frac{1}{x^{2}}$

Step 1.1.1.4.2

Combine $5$ and $\frac{1}{x^{2}}$ .

$2 + \frac{5}{x^{2}}$

Step 1.1.1.4.3

Reorder terms.

$f' (x) = \frac{5}{x^{2}} + 2$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{5}{x^{2}} + 2$ .

$\frac{5}{x^{2}} + 2$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{5}{x^{2}} + 2 = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$\frac{5}{x^{2}} + 2 = 0$

Step 1.2.2

Subtract $2$ from both sides of the equation.

$\frac{5}{x^{2}} = - 2$

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 $\frac{5}{x^{2}} = - 2$ by $x^{2}$ to eliminate the fractions.

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Step 1.2.4.1

Multiply each term in $\frac{5}{x^{2}} = - 2$ by $x^{2}$ .

$\frac{5}{x^{2}} x^{2} = - 2 x^{2}$

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

Cancel the common factor.

$\frac{5}{x^{2}} x^{2} = - 2 x^{2}$

Step 1.2.4.2.1.2

Rewrite the expression.

$5 = - 2 x^{2}$

Step 1.2.5

Solve the equation.

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Step 1.2.5.1

Rewrite the equation as $- 2 x^{2} = 5$ .

$- 2 x^{2} = 5$

Step 1.2.5.2

Divide each term in $- 2 x^{2} = 5$ by $- 2$ and simplify.

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Step 1.2.5.2.1

Divide each term in $- 2 x^{2} = 5$ by $- 2$ .

$\frac{- 2 x^{2}}{- 2} = \frac{5}{- 2}$

Step 1.2.5.2.2

Simplify the left side.

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Step 1.2.5.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.5.2.2.1.1

Cancel the common factor.

$\frac{- 2 x^{2}}{- 2} = \frac{5}{- 2}$

Step 1.2.5.2.2.1.2

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

$x^{2} = \frac{5}{- 2}$

Step 1.2.5.2.3

Simplify the right side.

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Step 1.2.5.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{5}{2}$

Step 1.2.5.3

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

$x = \pm \sqrt{- \frac{5}{2}}$

Step 1.2.5.4

Simplify $\pm \sqrt{- \frac{5}{2}}$ .

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Step 1.2.5.4.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{5}{2}}$

Step 1.2.5.4.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{5}{2}}$

Step 1.2.5.4.3

Rewrite $\sqrt{\frac{5}{2}}$ as $\frac{\sqrt{5}}{\sqrt{2}}$ .

$x = \pm i \frac{\sqrt{5}}{\sqrt{2}}$

Step 1.2.5.4.4

Multiply $\frac{\sqrt{5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i (\frac{\sqrt{5}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 1.2.5.4.5

Combine and simplify the denominator.

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Step 1.2.5.4.5.1

Multiply $\frac{\sqrt{5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i \frac{\sqrt{5} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.2.5.4.5.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.2.5.4.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.2.5.4.5.4

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

$x = \pm i \frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.2.5.4.5.5

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.2.5.4.5.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.2.5.4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{5} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.2.5.4.5.6.2

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

$x = \pm i \frac{\sqrt{5} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.2.5.4.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm i \frac{\sqrt{5} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.5.4.5.6.4

Cancel the common factor of $2$ .

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Step 1.2.5.4.5.6.4.1

Cancel the common factor.

$x = \pm i \frac{\sqrt{5} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.5.4.5.6.4.2

Rewrite the expression.

$x = \pm i \frac{\sqrt{5} \sqrt{2}}{2^{1}}$

Step 1.2.5.4.5.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{5} \sqrt{2}}{2}$

Step 1.2.5.4.6

Simplify the numerator.

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Step 1.2.5.4.6.1

Combine using the product rule for radicals.

$x = \pm i \frac{\sqrt{5 \cdot 2}}{2}$

Step 1.2.5.4.6.2

Multiply $5$ by $2$ .

$x = \pm i \frac{\sqrt{10}}{2}$

Step 1.2.5.4.7

Combine $i$ and $\frac{\sqrt{10}}{2}$ .

$x = \pm \frac{i \sqrt{10}}{2}$

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.

$x = \frac{i \sqrt{10}}{2}$

Step 1.2.5.5.2

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

$x = - \frac{i \sqrt{10}}{2}$

Step 1.2.5.5.3

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

$x = \frac{i \sqrt{10}}{2}, - \frac{i \sqrt{10}}{2}$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Set the denominator in $\frac{5}{x^{2}}$ 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 $2 x - \frac{5}{x}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 0$ .

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Step 1.4.1.1

Substitute $0$ for $x$ .

$2 (0) - \frac{5}{0}$

Step 1.4.1.2

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

Undefined

Step 1.5

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = 5$ .

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Step 2.1.1

Substitute $5$ for $x$ .

$2 (5) - \frac{5}{5}$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Multiply $2$ by $5$ .

$10 - \frac{5}{5}$

Step 2.1.2.1.2

Cancel the common factor of $5$ .

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Step 2.1.2.1.2.1

Cancel the common factor.

$10 - \frac{5}{5}$

Step 2.1.2.1.2.2

Rewrite the expression.

$10 - 1 \cdot 1$

Step 2.1.2.1.3

Multiply $- 1$ by $1$ .

$10 - 1$

Step 2.1.2.2

Subtract $1$ from $10$ .

$9$

Step 2.2

Evaluate at $x = 7$ .

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Step 2.2.1

Substitute $7$ for $x$ .

$2 (7) - \frac{5}{7}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Multiply $2$ by $7$ .

$14 - \frac{5}{7}$

Step 2.2.2.2

To write $14$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$14 \cdot \frac{7}{7} - \frac{5}{7}$

Step 2.2.2.3

Combine $14$ and $\frac{7}{7}$ .

$\frac{14 \cdot 7}{7} - \frac{5}{7}$

Step 2.2.2.4

Combine the numerators over the common denominator.

$\frac{14 \cdot 7 - 5}{7}$

Step 2.2.2.5

Simplify the numerator.

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Step 2.2.2.5.1

Multiply $14$ by $7$ .

$\frac{98 - 5}{7}$

Step 2.2.2.5.2

Subtract $5$ from $98$ .

$\frac{93}{7}$

Step 2.3

List all of the points.

$(5, 9), (7, \frac{93}{7})$

Step 3

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: $(7, \frac{93}{7})$

Absolute Minimum: $(5, 9)$

Step 4