Calculus Examples

$f (x) = 1 + \sqrt{x - 5}$ ; $6 \leq x \leq 9$

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 $1 + \sqrt{x - 5}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\sqrt{x - 5}]$ .

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

Step 1.1.1.1.2

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

$0 + \frac{d}{d x} [\sqrt{x - 5}]$

Step 1.1.1.2

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

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Step 1.1.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 5}$ as ${(x - 5)}^{\frac{1}{2}}$ .

$0 + \frac{d}{d x} [{(x - 5)}^{\frac{1}{2}}]$

Step 1.1.1.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x - 5$ .

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Step 1.1.1.2.2.1

To apply the Chain Rule, set $u$ as $x - 5$ .

$0 + \frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x - 5]$

Step 1.1.1.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{2}$ .

$0 + \frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x - 5]$

Step 1.1.1.2.2.3

Replace all occurrences of $u$ with $x - 5$ .

$0 + \frac{1}{2} {(x - 5)}^{\frac{1}{2} - 1} \frac{d}{d x} [x - 5]$

Step 1.1.1.2.3

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

$0 + \frac{1}{2} {(x - 5)}^{\frac{1}{2} - 1} (\frac{d}{d x} [x] + \frac{d}{d x} [- 5])$

Step 1.1.1.2.4

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

$0 + \frac{1}{2} {(x - 5)}^{\frac{1}{2} - 1} (1 + \frac{d}{d x} [- 5])$

Step 1.1.1.2.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$0 + \frac{1}{2} {(x - 5)}^{\frac{1}{2} - 1} (1 + 0)$

Step 1.1.1.2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$0 + \frac{1}{2} {(x - 5)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (1 + 0)$

Step 1.1.1.2.7

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

$0 + \frac{1}{2} {(x - 5)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (1 + 0)$

Step 1.1.1.2.8

Combine the numerators over the common denominator.

$0 + \frac{1}{2} {(x - 5)}^{\frac{1 - 1 \cdot 2}{2}} (1 + 0)$

Step 1.1.1.2.9

Simplify the numerator.

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Step 1.1.1.2.9.1

Multiply $- 1$ by $2$ .

$0 + \frac{1}{2} {(x - 5)}^{\frac{1 - 2}{2}} (1 + 0)$

Step 1.1.1.2.9.2

Subtract $2$ from $1$ .

$0 + \frac{1}{2} {(x - 5)}^{\frac{- 1}{2}} (1 + 0)$

Step 1.1.1.2.10

Move the negative in front of the fraction.

$0 + \frac{1}{2} {(x - 5)}^{- \frac{1}{2}} (1 + 0)$

Step 1.1.1.2.11

Add $1$ and $0$ .

$0 + \frac{1}{2} {(x - 5)}^{- \frac{1}{2}} \cdot 1$

Step 1.1.1.2.12

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

$0 + \frac{{(x - 5)}^{- \frac{1}{2}}}{2} \cdot 1$

Step 1.1.1.2.13

Multiply $\frac{{(x - 5)}^{- \frac{1}{2}}}{2}$ by $1$ .

$0 + \frac{{(x - 5)}^{- \frac{1}{2}}}{2}$

Step 1.1.1.2.14

Move ${(x - 5)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 + \frac{1}{2 {(x - 5)}^{\frac{1}{2}}}$

Step 1.1.1.3

Add $0$ and $\frac{1}{2 {(x - 5)}^{\frac{1}{2}}}$ .

$f' (x) = \frac{1}{2 {(x - 5)}^{\frac{1}{2}}}$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{1}{2 {(x - 5)}^{\frac{1}{2}}}$ .

$\frac{1}{2 {(x - 5)}^{\frac{1}{2}}}$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{1}{2 {(x - 5)}^{\frac{1}{2}}} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$\frac{1}{2 {(x - 5)}^{\frac{1}{2}}} = 0$

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

Convert expressions with fractional exponents to radicals.

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Step 1.3.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{1}{2 \sqrt{{(x - 5)}^{1}}}$

Step 1.3.1.2

Anything raised to $1$ is the base itself.

$\frac{1}{2 \sqrt{x - 5}}$

Step 1.3.2

Set the denominator in $\frac{1}{2 \sqrt{x - 5}}$ equal to $0$ to find where the expression is undefined.

$2 \sqrt{x - 5} = 0$

Step 1.3.3

Solve for $x$ .

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Step 1.3.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${(2 \sqrt{x - 5})}^{2} = 0^{2}$

Step 1.3.3.2

Simplify each side of the equation.

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Step 1.3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 5}$ as ${(x - 5)}^{\frac{1}{2}}$ .

${(2 {(x - 5)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

Simplify ${(2 {(x - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 1.3.3.2.2.1.1

Apply the product rule to $2 {(x - 5)}^{\frac{1}{2}}$ .

$2^{2} {({(x - 5)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.3.3.2.2.1.2

Raise $2$ to the power of $2$ .

$4 {({(x - 5)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.3.3.2.2.1.3

Multiply the exponents in ${({(x - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 1.3.3.2.2.1.3.1

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

$4 {(x - 5)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.3.3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 1.3.3.2.2.1.3.2.1

Cancel the common factor.

$4 {(x - 5)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.3.3.2.2.1.3.2.2

Rewrite the expression.

$4 {(x - 5)}^{1} = 0^{2}$

Step 1.3.3.2.2.1.4

Simplify.

$4 (x - 5) = 0^{2}$

Step 1.3.3.2.2.1.5

Apply the distributive property.

$4 x + 4 \cdot - 5 = 0^{2}$

Step 1.3.3.2.2.1.6

Multiply $4$ by $- 5$ .

$4 x - 20 = 0^{2}$

Step 1.3.3.2.3

Simplify the right side.

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Step 1.3.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$4 x - 20 = 0$

Step 1.3.3.3

Solve for $x$ .

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Step 1.3.3.3.1

Add $20$ to both sides of the equation.

$4 x = 20$

Step 1.3.3.3.2

Divide each term in $4 x = 20$ by $4$ and simplify.

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Step 1.3.3.3.2.1

Divide each term in $4 x = 20$ by $4$ .

$\frac{4 x}{4} = \frac{20}{4}$

Step 1.3.3.3.2.2

Simplify the left side.

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Step 1.3.3.3.2.2.1

Cancel the common factor of $4$ .

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Step 1.3.3.3.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{20}{4}$

Step 1.3.3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{20}{4}$

Step 1.3.3.3.2.3

Simplify the right side.

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Step 1.3.3.3.2.3.1

Divide $20$ by $4$ .

$x = 5$

Step 1.3.4

Set the radicand in $\sqrt{x - 5}$ less than $0$ to find where the expression is undefined.

$x - 5 < 0$

Step 1.3.5

Add $5$ to both sides of the inequality.

$x < 5$

Step 1.3.6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 5$

$(- \infty, 5]$

$x \leq 5$

$(- \infty, 5]$

Step 1.4

Evaluate $1 + \sqrt{x - 5}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 5$ .

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Step 1.4.1.1

Substitute $5$ for $x$ .

$1 + \sqrt{(5) - 5}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

Subtract $5$ from $5$ .

$1 + \sqrt{0}$

Step 1.4.1.2.1.2

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

$1 + \sqrt{0^{2}}$

Step 1.4.1.2.1.3

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

$1 + 0$

Step 1.4.1.2.2

Add $1$ and $0$ .

$1$

Step 1.4.2

List all of the points.

$(5, 1)$

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 $x = 6$ .

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Step 3.1.1

Substitute $6$ for $x$ .

$1 + \sqrt{(6) - 5}$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Subtract $5$ from $6$ .

$1 + \sqrt{1}$

Step 3.1.2.1.2

Any root of $1$ is $1$ .

$1 + 1$

Step 3.1.2.2

Add $1$ and $1$ .

$2$

Step 3.2

Evaluate at $x = 9$ .

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Step 3.2.1

Substitute $9$ for $x$ .

$1 + \sqrt{(9) - 5}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Subtract $5$ from $9$ .

$1 + \sqrt{4}$

Step 3.2.2.1.2

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

$1 + \sqrt{2^{2}}$

Step 3.2.2.1.3

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

$1 + 2$

Step 3.2.2.2

Add $1$ and $2$ .

$3$

Step 3.3

List all of the points.

$(6, 2), (9, 3)$

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: $(9, 3)$

Absolute Minimum: $(6, 2)$

Step 5