Calculus Examples

$f (x) = | x |$ , given $\sqrt{5} \leq x \leq \sqrt{13}$

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

The derivative of $| x |$ with respect to $x$ is $\frac{x}{| x |}$ .

$f' (x) = \frac{x}{| x |}$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x}{| x |}$ .

$\frac{x}{| x |}$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{x}{| x |} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$\frac{x}{| x |} = 0$

Step 1.2.2

Set the numerator equal to zero.

$x = 0$

Step 1.2.3

Exclude the solutions that do not make $\frac{x}{| x |} = 0$ true.

$No solution$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

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

$| x | = 0$

Step 1.3.2

Solve for $x$ .

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Step 1.3.2.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x = \pm 0$

Step 1.3.2.2

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.4

Evaluate $| x |$ 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 $x$ .

$| 0 |$

Step 1.4.1.2

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$0$

Step 1.4.2

List all of the points.

$(0, 0)$

Step 2

Exclude the points that are not on the interval.

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = \sqrt{5}$ .

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Step 3.1.1

Substitute $\sqrt{5}$ for $x$ .

$| \sqrt{5} |$

Step 3.1.2

$\sqrt{5}$ is approximately $2.23606797$ which is positive so remove the absolute value

$\sqrt{5}$

Step 3.2

Evaluate at $x = \sqrt{13}$ .

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Step 3.2.1

Substitute $\sqrt{13}$ for $x$ .

$| \sqrt{13} |$

Step 3.2.2

$\sqrt{13}$ is approximately $3.60555127$ which is positive so remove the absolute value

$\sqrt{13}$

Step 3.3

List all of the points.

$(\sqrt{5}, \sqrt{5}), (\sqrt{13}, \sqrt{13})$

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: $(\sqrt{13}, \sqrt{13})$

Absolute Minimum: $(\sqrt{5}, \sqrt{5})$

Step 5