Calculus Examples

$f (x) = \frac{1}{x}$ , $x \geq 1$

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

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

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

Step 1.1.1.2

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

$- x^{- 2}$

Step 1.1.1.3

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

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

Set the first derivative equal to $0$ .

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

Step 1.2.2

Set the numerator equal to zero.

$1 = 0$

Step 1.2.3

Since $1 \neq 0$ , there are no solutions.

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 $\frac{1}{x^{2}}$ 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 $\frac{1}{x}$ 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 $x$ .

$\frac{1}{0}$

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 $x = 1$ .

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Step 2.1.1

Substitute $1$ for $x$ .

$\frac{1}{1}$

Step 2.1.2

Divide $1$ by $1$ .

$1$

Step 2.2

List all of the points.

$(1, 1)$

Step 3

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

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: $(1, 1)$

No absolute minimum

Step 5