Calculus Examples

$f (x) = \frac{9}{x - 1}$ , $[1, 4]$

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 using the Constant Multiple Rule.

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Step 1.1.1.1.1

Since $9$ is constant with respect to $x$ , the derivative of $\frac{9}{x - 1}$ with respect to $x$ is $9 \frac{d}{d x} [\frac{1}{x - 1}]$ .

$9 \frac{d}{d x} [\frac{1}{x - 1}]$

Step 1.1.1.1.2

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

$9 \frac{d}{d x} [{(x - 1)}^{- 1}]$

Step 1.1.1.2

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 $g (x) = x - 1$ .

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Step 1.1.1.2.1

To apply the Chain Rule, set $u$ as $x - 1$ .

$9 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x - 1])$

Step 1.1.1.2.2

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

$9 (- u^{- 2} \frac{d}{d x} [x - 1])$

Step 1.1.1.2.3

Replace all occurrences of $u$ with $x - 1$ .

$9 (- {(x - 1)}^{- 2} \frac{d}{d x} [x - 1])$

Step 1.1.1.3

Differentiate.

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Step 1.1.1.3.1

Multiply $- 1$ by $9$ .

$- 9 ({(x - 1)}^{- 2} \frac{d}{d x} [x - 1])$

Step 1.1.1.3.2

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$- 9 {(x - 1)}^{- 2} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

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 $n = 1$ .

$- 9 {(x - 1)}^{- 2} (1 + \frac{d}{d x} [- 1])$

Step 1.1.1.3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$- 9 {(x - 1)}^{- 2} (1 + 0)$

Step 1.1.1.3.5

Simplify the expression.

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Step 1.1.1.3.5.1

Add $1$ and $0$ .

$- 9 {(x - 1)}^{- 2} \cdot 1$

Step 1.1.1.3.5.2

Multiply $- 9$ by $1$ .

$- 9 {(x - 1)}^{- 2}$

Step 1.1.1.4

Simplify.

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Step 1.1.1.4.1

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

$- 9 \frac{1}{{(x - 1)}^{2}}$

Step 1.1.1.4.2

Combine terms.

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Step 1.1.1.4.2.1

Combine $- 9$ and $\frac{1}{{(x - 1)}^{2}}$ .

$\frac{- 9}{{(x - 1)}^{2}}$

Step 1.1.1.4.2.2

Move the negative in front of the fraction.

$f' (x) = - \frac{9}{{(x - 1)}^{2}}$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{9}{{(x - 1)}^{2}}$ .

$- \frac{9}{{(x - 1)}^{2}}$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $- \frac{9}{{(x - 1)}^{2}} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$- \frac{9}{{(x - 1)}^{2}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$9 = 0$

Step 1.2.3

Since $9 \neq 0$ , 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 $\frac{9}{{(x - 1)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x - 1)}^{2} = 0$

Step 1.3.2

Solve for $x$ .

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Step 1.3.2.1

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 1.3.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.4

Evaluate $\frac{9}{x - 1}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 1$ .

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Step 1.4.1.1

Substitute $1$ for $x$ .

$\frac{9}{(1) - 1}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Subtract $1$ from $1$ .

$\frac{9}{0}$

Step 1.4.1.2.2

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 $x$ .

$\frac{9}{(1) - 1}$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Subtract $1$ from $1$ .

$\frac{9}{0}$

Step 2.1.2.2

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

Undefined

Step 2.2

Evaluate at $x = 4$ .

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Step 2.2.1

Substitute $4$ for $x$ .

$\frac{9}{(4) - 1}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Subtract $1$ from $4$ .

$\frac{9}{3}$

Step 2.2.2.2

Divide $9$ by $3$ .

$3$

Step 2.3

List all of the points.

$(4, 3)$

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.

No absolute maximum

Absolute Minimum: $(4, 3)$

Step 5