Calculus Examples

f (x) = - \sqrt{x - 3}

$f (x) = - \sqrt{x - 3}$ ;

4 \leq x \leq 12

$4 \leq x \leq 12$

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

Use

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

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

\sqrt{x - 3}

$\sqrt{x - 3}$ as

{(x - 3)}^{\frac{1}{2}}

${(x - 3)}^{\frac{1}{2}}$ .

\frac{d}{d x} [- {(x - 3)}^{\frac{1}{2}}]

$\frac{d}{d x} [- {(x - 3)}^{\frac{1}{2}}]$

Step 1.1.1.1.2

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- {(x - 3)}^{\frac{1}{2}}

$- {(x - 3)}^{\frac{1}{2}}$ with respect to

x

$x$ is

- \frac{d}{d x} [{(x - 3)}^{\frac{1}{2}}]

$- \frac{d}{d x} [{(x - 3)}^{\frac{1}{2}}]$ .

- \frac{d}{d x} [{(x - 3)}^{\frac{1}{2}}]

$- \frac{d}{d x} [{(x - 3)}^{\frac{1}{2}}]$

- \frac{d}{d x} [{(x - 3)}^{\frac{1}{2}}]

$- \frac{d}{d x} [{(x - 3)}^{\frac{1}{2}}]$

Step 1.1.1.2

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 - 3

$g (x) = x - 3$ .

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Step 1.1.1.2.1

To apply the Chain Rule, set

u

$u$ as

x - 3

$x - 3$ .

- (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x - 3])

$- (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x - 3])$

Step 1.1.1.2.2

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 - 3])

$- (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x - 3])$

Step 1.1.1.2.3

Replace all occurrences of

u

$u$ with

x - 3

$x - 3$ .

- (\frac{1}{2} {(x - 3)}^{\frac{1}{2} - 1} \frac{d}{d x} [x - 3])

$- (\frac{1}{2} {(x - 3)}^{\frac{1}{2} - 1} \frac{d}{d x} [x - 3])$

- (\frac{1}{2} {(x - 3)}^{\frac{1}{2} - 1} \frac{d}{d x} [x - 3])

$- (\frac{1}{2} {(x - 3)}^{\frac{1}{2} - 1} \frac{d}{d x} [x - 3])$

Step 1.1.1.3

To write

- 1

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

\frac{2}{2}

$\frac{2}{2}$ .

- (\frac{1}{2} {(x - 3)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x - 3])

$- (\frac{1}{2} {(x - 3)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x - 3])$

Step 1.1.1.4

Combine

- 1

$- 1$ and

\frac{2}{2}

$\frac{2}{2}$ .

- (\frac{1}{2} {(x - 3)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x - 3])

$- (\frac{1}{2} {(x - 3)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x - 3])$

Step 1.1.1.5

Combine the numerators over the common denominator.

- (\frac{1}{2} {(x - 3)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x - 3])

$- (\frac{1}{2} {(x - 3)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x - 3])$

Step 1.1.1.6

Simplify the numerator.

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Step 1.1.1.6.1

Multiply

- 1

$- 1$ by

2

$2$ .

- (\frac{1}{2} {(x - 3)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x - 3])

$- (\frac{1}{2} {(x - 3)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x - 3])$

Step 1.1.1.6.2

Subtract

2

$2$ from

1

$1$ .

- (\frac{1}{2} {(x - 3)}^{\frac{- 1}{2}} \frac{d}{d x} [x - 3])

$- (\frac{1}{2} {(x - 3)}^{\frac{- 1}{2}} \frac{d}{d x} [x - 3])$

- (\frac{1}{2} {(x - 3)}^{\frac{- 1}{2}} \frac{d}{d x} [x - 3])

$- (\frac{1}{2} {(x - 3)}^{\frac{- 1}{2}} \frac{d}{d x} [x - 3])$

Step 1.1.1.7

Combine fractions.

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Step 1.1.1.7.1

Move the negative in front of the fraction.

- (\frac{1}{2} {(x - 3)}^{- \frac{1}{2}} \frac{d}{d x} [x - 3])

$- (\frac{1}{2} {(x - 3)}^{- \frac{1}{2}} \frac{d}{d x} [x - 3])$

Step 1.1.1.7.2

Combine

\frac{1}{2}

$\frac{1}{2}$ and

{(x - 3)}^{- \frac{1}{2}}

${(x - 3)}^{- \frac{1}{2}}$ .

- (\frac{{(x - 3)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x - 3])

$- (\frac{{(x - 3)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x - 3])$

Step 1.1.1.7.3

Move

{(x - 3)}^{- \frac{1}{2}}

${(x - 3)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule

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

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

- (\frac{1}{2 {(x - 3)}^{\frac{1}{2}}} \frac{d}{d x} [x - 3])

$- (\frac{1}{2 {(x - 3)}^{\frac{1}{2}}} \frac{d}{d x} [x - 3])$

- (\frac{1}{2 {(x - 3)}^{\frac{1}{2}}} \frac{d}{d x} [x - 3])

$- (\frac{1}{2 {(x - 3)}^{\frac{1}{2}}} \frac{d}{d x} [x - 3])$

Step 1.1.1.8

By the Sum Rule, the derivative of

x - 3

$x - 3$ with respect to

x

$x$ is

\frac{d}{d x} [x] + \frac{d}{d x} [- 3]

$\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 3])

$- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 3])$

Step 1.1.1.9

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 - 3)}^{\frac{1}{2}}} (1 + \frac{d}{d x} [- 3])

$- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}} (1 + \frac{d}{d x} [- 3])$

Step 1.1.1.10

Since

- 3

$- 3$ is constant with respect to

x

$x$ , the derivative of

- 3

$- 3$ with respect to

x

$x$ is

0

$0$ .

- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}} (1 + 0)

$- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}} (1 + 0)$

Step 1.1.1.11

Simplify the expression.

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Step 1.1.1.11.1

Add

1

$1$ and

0

$0$ .

- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}} \cdot 1

$- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}} \cdot 1$

Step 1.1.1.11.2

Multiply

- 1

$- 1$ by

1

$1$ .

f' (x) = - \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}

$f' (x) = - \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}$

f' (x) = - \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}

$f' (x) = - \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}$

f' (x) = - \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}

$f' (x) = - \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}$

Step 1.1.2

The first derivative of

f (x)

$f (x)$ with respect to

x

$x$ is

- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}

$- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}$ .

- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}

$- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}$

- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}

$- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}}$

Step 1.2

Set the first derivative equal to

0

$0$ then solve the equation

- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}} = 0

$- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}} = 0$ .

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Step 1.2.1

Set the first derivative equal to

0

$0$ .

- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}} = 0

$- \frac{1}{2 {(x - 3)}^{\frac{1}{2}}} = 0$

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 - 3)}^{1}}}

$- \frac{1}{2 \sqrt{{(x - 3)}^{1}}}$

Step 1.3.1.2

Anything raised to

1

$1$ is the base itself.

- \frac{1}{2 \sqrt{x - 3}}

$- \frac{1}{2 \sqrt{x - 3}}$

- \frac{1}{2 \sqrt{x - 3}}

$- \frac{1}{2 \sqrt{x - 3}}$

Step 1.3.2

Set the denominator in

\frac{1}{2 \sqrt{x - 3}}

$\frac{1}{2 \sqrt{x - 3}}$ equal to

0

$0$ to find where the expression is undefined.

2 \sqrt{x - 3} = 0

$2 \sqrt{x - 3} = 0$

Step 1.3.3

Solve for

x

$x$ .

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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 - 3})}^{2} = 0^{2}

${(2 \sqrt{x - 3})}^{2} = 0^{2}$

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}}

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

\sqrt{x - 3}

$\sqrt{x - 3}$ as

{(x - 3)}^{\frac{1}{2}}

${(x - 3)}^{\frac{1}{2}}$ .

${(2 {(x - 3)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

Simplify

${(2 {(x - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 1.3.3.2.2.1.1

Apply the product rule to

$2 {(x - 3)}^{\frac{1}{2}}$ .

$2^{2} {({(x - 3)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.3.3.2.2.1.2

Raise

$2$ to the power of

$2$ .

$4 {({(x - 3)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.3.3.2.2.1.3

Multiply the exponents in

${({(x - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 1.3.3.2.2.1.3.1

Apply the power rule and multiply exponents,

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

$4 {(x - 3)}^{\frac{1}{2} \cdot 2} = 0^{2}$

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.

$4 {(x - 3)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.3.3.2.2.1.3.2.2

Rewrite the expression.

$4 {(x - 3)}^{1} = 0^{2}$

Step 1.3.3.2.2.1.4

Simplify.

$4 (x - 3) = 0^{2}$

Step 1.3.3.2.2.1.5

Apply the distributive property.

$4 x + 4 \cdot - 3 = 0^{2}$

Step 1.3.3.2.2.1.6

Multiply

$4$ by

$- 3$ .

$4 x - 12 = 0^{2}$

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

$0$ .

$4 x - 12 = 0$

Step 1.3.3.3

Solve for

$x$ .

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Step 1.3.3.3.1

Add

$12$ to both sides of the equation.

$4 x = 12$

Step 1.3.3.3.2

Divide each term in

$4 x = 12$ by

$4$ and simplify.

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Step 1.3.3.3.2.1

Divide each term in

$4 x = 12$ by

$4$ .

$\frac{4 x}{4} = \frac{12}{4}$

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.

$\frac{4 x}{4} = \frac{12}{4}$

Step 1.3.3.3.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{12}{4}$

Step 1.3.3.3.2.3

Simplify the right side.

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Step 1.3.3.3.2.3.1

Divide

$12$ by

$4$ .

$x = 3$

Step 1.3.4

Set the radicand in

$\sqrt{x - 3}$ less than

$0$ to find where the expression is undefined.

$x - 3 < 0$

Step 1.3.5

Add

$3$ to both sides of the inequality.

$x < 3$

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

$0$ .

$x \leq 3$

$(- \infty, 3]$

$x \leq 3$

$(- \infty, 3]$

Step 1.4

Evaluate

$- \sqrt{x - 3}$ at each

$x$ value where the derivative is

$0$ or undefined.

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Step 1.4.1

Evaluate at

$x = 3$ .

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Step 1.4.1.1

Substitute

$3$ for

$x$ .

$- \sqrt{(3) - 3}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Subtract

$3$ from

$3$ .

$- \sqrt{0}$

Step 1.4.1.2.2

Rewrite

$0$ as

$0^{2}$ .

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

Step 1.4.1.2.3

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

$- 0$

Step 1.4.1.2.4

Multiply

$- 1$ by

$0$ .

$0$

Step 1.4.2

List all of the points.

$(3, 0)$

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

$x = 4$ .

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Step 3.1.1

Substitute

$4$ for

$x$ .

$- \sqrt{(4) - 3}$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Subtract

$3$ from

$4$ .

$- \sqrt{1}$

Step 3.1.2.2

Any root of

$1$ is

$1$ .

$- 1 \cdot 1$

Step 3.1.2.3

Multiply

$- 1$ by

$1$ .

$- 1$

Step 3.2

Evaluate at

$x = 12$ .

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Step 3.2.1

Substitute

$12$ for

$x$ .

$- \sqrt{(12) - 3}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Subtract

$3$ from

$12$ .

$- \sqrt{9}$

Step 3.2.2.2

Rewrite

$9$ as

$3^{2}$ .

$- \sqrt{3^{2}}$

Step 3.2.2.3

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

$- 1 \cdot 3$

Step 3.2.2.4

Multiply

$- 1$ by

$3$ .

$- 3$

Step 3.3

List all of the points.

$(4, - 1), (12, - 3)$

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:

$(4, - 1)$

Absolute Minimum:

$(12, - 3)$

Step 5