Calculus Examples

f (x) = - \sqrt{x + 1} + 2

$f (x) = - \sqrt{x + 1} + 2$ ;

3 \leq x \leq 10

$3 \leq x \leq 10$

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

- \sqrt{x + 1} + 2

$- \sqrt{x + 1} + 2$ with respect to

x

$x$ is

\frac{d}{d x} [- \sqrt{x + 1}] + \frac{d}{d x} [2]

$\frac{d}{d x} [- \sqrt{x + 1}] + \frac{d}{d x} [2]$ .

\frac{d}{d x} [- \sqrt{x + 1}] + \frac{d}{d x} [2]

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

Step 1.1.1.2

Evaluate

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

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

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Step 1.1.1.2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x + 1}

$\sqrt{x + 1}$ as

{(x + 1)}^{\frac{1}{2}}

${(x + 1)}^{\frac{1}{2}}$ .

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

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

Step 1.1.1.2.2

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

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

$- {(x + 1)}^{\frac{1}{2}}$ with respect to

x

$x$ is

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

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

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

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

Step 1.1.1.2.3

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

f' (g (x)) g' (x)

$f' (g (x)) g' (x)$ where

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

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

g (x) = x + 1

$g (x) = x + 1$ .

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Step 1.1.1.2.3.1

To apply the Chain Rule, set

u

$u$ as

x + 1

$x + 1$ .

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

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

Step 1.1.1.2.3.2

Differentiate using the Power Rule which states that

\frac{d}{d u} [u^{n}]

$\frac{d}{d u} [u^{n}]$ is

n u^{n - 1}

$n u^{n - 1}$ where

n = \frac{1}{2}

$n = \frac{1}{2}$ .

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

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

Step 1.1.1.2.3.3

Replace all occurrences of

u

$u$ with

x + 1

$x + 1$ .

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

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

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

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

Step 1.1.1.2.4

By the Sum Rule, the derivative of

x + 1

$x + 1$ with respect to

x

$x$ is

\frac{d}{d x} [x] + \frac{d}{d x} [1]

$\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

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

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

Step 1.1.1.2.5

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

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

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

Step 1.1.1.2.6

Since

1

$1$ is constant with respect to

x

$x$ , the derivative of

1

$1$ with respect to

x

$x$ is

0

$0$ .

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

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

Step 1.1.1.2.7

To write

- 1

$- 1$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 1.1.1.2.8

Combine

- 1

$- 1$ and

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 1.1.1.2.9

Combine the numerators over the common denominator.

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

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

Step 1.1.1.2.10

Simplify the numerator.

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Step 1.1.1.2.10.1

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

Step 1.1.1.2.10.2

Subtract

2

$2$ from

1

$1$ .

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

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

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

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

Step 1.1.1.2.11

Move the negative in front of the fraction.

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

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

Step 1.1.1.2.12

Add

1

$1$ and

0

$0$ .

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

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

Step 1.1.1.2.13

Combine

\frac{1}{2}

$\frac{1}{2}$ and

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

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

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

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

Step 1.1.1.2.14

Multiply

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

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

1

$1$ .

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

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

Step 1.1.1.2.15

Move

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

${(x + 1)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

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

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

Step 1.1.1.3

Differentiate using the Constant Rule.

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Step 1.1.1.3.1

Since

2

$2$ is constant with respect to

x

$x$ , the derivative of

2

$2$ with respect to

x

$x$ is

0

$0$ .

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

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

Step 1.1.1.3.2

Add

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

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

0

$0$ .

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

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

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

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

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

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

Step 1.1.2

The first derivative of

f (x)

$f (x)$ with respect to

x

$x$ is

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

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

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

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

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

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

Step 1.2

Set the first derivative equal to

0

$0$ then solve the equation

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

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

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Step 1.2.1

Set the first derivative equal to

0

$0$ .

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

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

Step 1.2.2

Set the numerator equal to zero.

1 = 0

$1 = 0$

Step 1.2.3

Since

1 \neq 0

$1 \neq 0$ , 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}}

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

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

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

Step 1.3.1.2

Anything raised to

1

$1$ is the base itself.

- \frac{1}{2 \sqrt{x + 1}}

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

- \frac{1}{2 \sqrt{x + 1}}

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

Step 1.3.2

Set the denominator in

\frac{1}{2 \sqrt{x + 1}}

$\frac{1}{2 \sqrt{x + 1}}$ equal to

$0$ to find where the expression is undefined.

$2 \sqrt{x + 1} = 0$

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.

${(2 \sqrt{x + 1})}^{2} = 0^{2}$

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

$\sqrt{x + 1}$ as

${(x + 1)}^{\frac{1}{2}}$ .

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

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

Simplify

${(2 {(x + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 1.3.3.2.2.1.1

Apply the product rule to

$2 {(x + 1)}^{\frac{1}{2}}$ .

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

Step 1.3.3.2.2.1.2

Raise

$2$ to the power of

$2$ .

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

Step 1.3.3.2.2.1.3

Multiply the exponents in

${({(x + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 1.3.3.2.2.1.3.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

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.

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

Step 1.3.3.2.2.1.3.2.2

Rewrite the expression.

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

Step 1.3.3.2.2.1.4

Simplify.

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

Step 1.3.3.2.2.1.5

Apply the distributive property.

$4 x + 4 \cdot 1 = 0^{2}$

Step 1.3.3.2.2.1.6

Multiply

$4$ by

$1$ .

$4 x + 4 = 0^{2}$

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

$0$ .

$4 x + 4 = 0$

Step 1.3.3.3

Solve for

$x$ .

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Step 1.3.3.3.1

Subtract

$4$ from both sides of the equation.

$4 x = - 4$

Step 1.3.3.3.2

Divide each term in

$4 x = - 4$ by

$4$ and simplify.

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Step 1.3.3.3.2.1

Divide each term in

$4 x = - 4$ by

$4$ .

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

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.

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

Step 1.3.3.3.2.2.1.2

Divide

$x$ by

$1$ .

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

Step 1.3.3.3.2.3

Simplify the right side.

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Step 1.3.3.3.2.3.1

Divide

$- 4$ by

$4$ .

$x = - 1$

Step 1.3.4

Set the radicand in

$\sqrt{x + 1}$ less than

$0$ to find where the expression is undefined.

$x + 1 < 0$

Step 1.3.5

Subtract

$1$ from both sides of the inequality.

$x < - 1$

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

$0$ .

$x \leq - 1$

$(- \infty, - 1]$

$x \leq - 1$

$(- \infty, - 1]$

Step 1.4

Evaluate

$- \sqrt{x + 1} + 2$ 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

$x$ .

$- \sqrt{(- 1) + 1} + 2$

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

Add

$- 1$ and

$1$ .

$- \sqrt{0} + 2$

Step 1.4.1.2.1.2

Rewrite

$0$ as

$0^{2}$ .

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

Step 1.4.1.2.1.3

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

$- 0 + 2$

Step 1.4.1.2.1.4

Multiply

$- 1$ by

$0$ .

$0 + 2$

Step 1.4.1.2.2

Add

$0$ and

$2$ .

$2$

Step 1.4.2

List all of the points.

$(- 1, 2)$

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

$x = 3$ .

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Step 3.1.1

Substitute

$3$ for

$x$ .

$- \sqrt{(3) + 1} + 2$

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

Add

$3$ and

$1$ .

$- \sqrt{4} + 2$

Step 3.1.2.1.2

Rewrite

$4$ as

$2^{2}$ .

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

Step 3.1.2.1.3

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

$- 1 \cdot 2 + 2$

Step 3.1.2.1.4

Multiply

$- 1$ by

$2$ .

$- 2 + 2$

Step 3.1.2.2

Add

$- 2$ and

$2$ .

$0$

Step 3.2

Evaluate at

$x = 10$ .

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Step 3.2.1

Substitute

$10$ for

$x$ .

$- \sqrt{(10) + 1} + 2$

Step 3.2.2

Add

$10$ and

$1$ .

$- \sqrt{11} + 2$

Step 3.3

List all of the points.

$(3, 0), (10, - \sqrt{11} + 2)$

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:

$(3, 0)$

Absolute Minimum:

$(10, - \sqrt{11} + 2)$

Step 5