Calculus Examples

g (x) = - \sqrt{4 - x^{2}}

$g (x) = - \sqrt{4 - x^{2}}$ ,

- 2 \leq x \leq 1

$- 2 \leq x \leq 1$

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

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{4 - x^{2}}

$\sqrt{4 - x^{2}}$ as

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

${(4 - x^{2})}^{\frac{1}{2}}$ .

\frac{d}{d x} [- {(4 - x^{2})}^{\frac{1}{2}}]

$\frac{d}{d x} [- {(4 - x^{2})}^{\frac{1}{2}}]$

Step 1.1.1.1.2

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

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

$- {(4 - x^{2})}^{\frac{1}{2}}$ with respect to

x

$x$ is

- \frac{d}{d x} [{(4 - x^{2})}^{\frac{1}{2}}]

$- \frac{d}{d x} [{(4 - x^{2})}^{\frac{1}{2}}]$ .

- \frac{d}{d x} [{(4 - x^{2})}^{\frac{1}{2}}]

$- \frac{d}{d x} [{(4 - x^{2})}^{\frac{1}{2}}]$

- \frac{d}{d x} [{(4 - x^{2})}^{\frac{1}{2}}]

$- \frac{d}{d x} [{(4 - x^{2})}^{\frac{1}{2}}]$

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

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

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

$g (x) = 4 - x^{2}$ .

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Step 1.1.1.2.1

To apply the Chain Rule, set

$u$ as

$4 - x^{2}$ .

$- (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [4 - x^{2}])$

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

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

$- (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [4 - x^{2}])$

Step 1.1.1.2.3

Replace all occurrences of

$u$ with

$4 - x^{2}$ .

$- (\frac{1}{2} {(4 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [4 - x^{2}])$

Step 1.1.1.3

To write

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

$\frac{2}{2}$ .

$- (\frac{1}{2} {(4 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [4 - x^{2}])$

Step 1.1.1.4

Combine

$- 1$ and

$\frac{2}{2}$ .

$- (\frac{1}{2} {(4 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [4 - x^{2}])$

Step 1.1.1.5

Combine the numerators over the common denominator.

$- (\frac{1}{2} {(4 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [4 - x^{2}])$

Step 1.1.1.6

Simplify the numerator.

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Step 1.1.1.6.1

Multiply

$- 1$ by

$2$ .

$- (\frac{1}{2} {(4 - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [4 - x^{2}])$

Step 1.1.1.6.2

Subtract

$2$ from

$1$ .

$- (\frac{1}{2} {(4 - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} [4 - x^{2}])$

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.

$- (\frac{1}{2} {(4 - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} [4 - x^{2}])$

Step 1.1.1.7.2

Combine

$\frac{1}{2}$ and

${(4 - x^{2})}^{- \frac{1}{2}}$ .

$- (\frac{{(4 - x^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [4 - x^{2}])$

Step 1.1.1.7.3

Move

${(4 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule

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

$- (\frac{1}{2 {(4 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [4 - x^{2}])$

Step 1.1.1.8

By the Sum Rule, the derivative of

$4 - x^{2}$ with respect to

$x$ is

$\frac{d}{d x} [4] + \frac{d}{d x} [- x^{2}]$ .

$- \frac{1}{2 {(4 - x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [4] + \frac{d}{d x} [- x^{2}])$

Step 1.1.1.9

Since

$4$ is constant with respect to

$x$ , the derivative of

$4$ with respect to

$x$ is

$0$ .

$- \frac{1}{2 {(4 - x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x^{2}])$

Step 1.1.1.10

Add

$0$ and

$\frac{d}{d x} [- x^{2}]$ .

$- \frac{1}{2 {(4 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- x^{2}]$

Step 1.1.1.11

Since

$- 1$ is constant with respect to

$x$ , the derivative of

$- x^{2}$ with respect to

$x$ is

$- \frac{d}{d x} [x^{2}]$ .

$- \frac{1}{2 {(4 - x^{2})}^{\frac{1}{2}}} (- \frac{d}{d x} [x^{2}])$

Step 1.1.1.12

Multiply.

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Step 1.1.1.12.1

Multiply

$- 1$ by

$- 1$ .

$1 \frac{1}{2 {(4 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [x^{2}]$

Step 1.1.1.12.2

Multiply

$\frac{1}{2 {(4 - x^{2})}^{\frac{1}{2}}}$ by

$1$ .

$\frac{1}{2 {(4 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [x^{2}]$

Step 1.1.1.13

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 2$ .

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

Step 1.1.1.14

Simplify terms.

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Step 1.1.1.14.1

Combine

$2$ and

$\frac{1}{2 {(4 - x^{2})}^{\frac{1}{2}}}$ .

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

Step 1.1.1.14.2

Combine

$\frac{2}{2 {(4 - x^{2})}^{\frac{1}{2}}}$ and

$x$ .

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

Step 1.1.1.14.3

Cancel the common factor.

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

Step 1.1.1.14.4

Rewrite the expression.

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

Step 1.1.1.14.5

Reorder terms.

$f' (x) = \frac{x}{{(- x^{2} + 4)}^{\frac{1}{2}}}$

Step 1.1.2

The first derivative of

$g (x)$ with respect to

$x$ is

$\frac{x}{{(- x^{2} + 4)}^{\frac{1}{2}}}$ .

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

Step 1.2

Set the first derivative equal to

$0$ then solve the equation

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

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Step 1.2.1

Set the first derivative equal to

$0$ .

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

Step 1.2.2

Set the numerator equal to zero.

$x = 0$

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.

$\frac{x}{\sqrt{{(- x^{2} + 4)}^{1}}}$

Step 1.3.1.2

Anything raised to

$1$ is the base itself.

$\frac{x}{\sqrt{- x^{2} + 4}}$

Step 1.3.2

Set the denominator in

$\frac{x}{\sqrt{- x^{2} + 4}}$ equal to

$0$ to find where the expression is undefined.

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

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.

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

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

$\sqrt{- x^{2} + 4}$ as

${(- x^{2} + 4)}^{\frac{1}{2}}$ .

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

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

Simplify

${({(- x^{2} + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 1.3.3.2.2.1.1

Multiply the exponents in

${({(- x^{2} + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 1.3.3.2.2.1.1.1

Apply the power rule and multiply exponents,

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

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

Step 1.3.3.2.2.1.1.2

Cancel the common factor of

$2$ .

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Step 1.3.3.2.2.1.1.2.1

Cancel the common factor.

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

Step 1.3.3.2.2.1.1.2.2

Rewrite the expression.

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

Step 1.3.3.2.2.1.2

Simplify.

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

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

$0$ .

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

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.

$- x^{2} = - 4$

Step 1.3.3.3.2

Divide each term in

$- x^{2} = - 4$ by

$- 1$ and simplify.

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Step 1.3.3.3.2.1

Divide each term in

$- x^{2} = - 4$ by

$- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 4}{- 1}$

Step 1.3.3.3.2.2

Simplify the left side.

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Step 1.3.3.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 4}{- 1}$

Step 1.3.3.3.2.2.2

Divide

$x^{2}$ by

$1$ .

$x^{2} = \frac{- 4}{- 1}$

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

$- 1$ .

$x^{2} = 4$

Step 1.3.3.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Step 1.3.3.3.4

Simplify

$\pm \sqrt{4}$ .

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Step 1.3.3.3.4.1

Rewrite

$4$ as

$2^{2}$ .

$x = \pm \sqrt{2^{2}}$

Step 1.3.3.3.4.2

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

$x = \pm 2$

Step 1.3.3.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.3.3.3.5.1

First, use the positive value of the

$\pm$ to find the first solution.

$x = 2$

Step 1.3.3.3.5.2

Next, use the negative value of the

$\pm$ to find the second solution.

$x = - 2$

Step 1.3.3.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

Step 1.3.4

Set the radicand in

$\sqrt{- x^{2} + 4}$ less than

$0$ to find where the expression is undefined.

$- x^{2} + 4 < 0$

Step 1.3.5

Solve for

$x$ .

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Step 1.3.5.1

Subtract

$4$ from both sides of the inequality.

$- x^{2} < - 4$

Step 1.3.5.2

Divide each term in

$- x^{2} < - 4$ by

$- 1$ and simplify.

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Step 1.3.5.2.1

Divide each term in

$- x^{2} < - 4$ by

$- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} > \frac{- 4}{- 1}$

Step 1.3.5.2.2

Simplify the left side.

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Step 1.3.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} > \frac{- 4}{- 1}$

Step 1.3.5.2.2.2

Divide

$x^{2}$ by

$1$ .

$x^{2} > \frac{- 4}{- 1}$

Step 1.3.5.2.3

Simplify the right side.

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Step 1.3.5.2.3.1

Divide

$- 4$ by

$- 1$ .

$x^{2} > 4$

Step 1.3.5.3

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} > \sqrt{4}$

Step 1.3.5.4

Simplify the equation.

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Step 1.3.5.4.1

Simplify the left side.

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Step 1.3.5.4.1.1

Pull terms out from under the radical.

$| x | > \sqrt{4}$

Step 1.3.5.4.2

Simplify the right side.

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Step 1.3.5.4.2.1

Simplify

$\sqrt{4}$ .

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Step 1.3.5.4.2.1.1

Rewrite

$4$ as

$2^{2}$ .

$| x | > \sqrt{2^{2}}$

Step 1.3.5.4.2.1.2

Pull terms out from under the radical.

$| x | > | 2 |$

Step 1.3.5.4.2.1.3

The absolute value is the distance between a number and zero. The distance between

$0$ and

$2$ is

$2$ .

$| x | > 2$

Step 1.3.5.5

Write

$| x | > 2$ as a piecewise.

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Step 1.3.5.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 1.3.5.5.2

In the piece where

$x$ is non-negative, remove the absolute value.

$x > 2$

Step 1.3.5.5.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 1.3.5.5.4

In the piece where

$x$ is negative, remove the absolute value and multiply by

$- 1$ .

$- x > 2$

Step 1.3.5.5.5

Write as a piecewise.

${\begin{cases} x > 2 & x \geq 0 \\ - x > 2 & x < 0 \end{cases}$

Step 1.3.5.6

Find the intersection of

$x > 2$ and

$x \geq 0$ .

$x > 2$

Step 1.3.5.7

Divide each term in

$- x > 2$ by

$- 1$ and simplify.

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Step 1.3.5.7.1

Divide each term in

$- x > 2$ by

$- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 1.3.5.7.2

Simplify the left side.

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Step 1.3.5.7.2.1

Dividing two negative values results in a positive value.

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

Step 1.3.5.7.2.2

Divide

$x$ by

$1$ .

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

Step 1.3.5.7.3

Simplify the right side.

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Step 1.3.5.7.3.1

Divide

$2$ by

$- 1$ .

$x < - 2$

Step 1.3.5.8

Find the union of the solutions.

$x < - 2$ or

$x > 2$

$x < - 2$ or

$x > 2$

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

$0$ .

$x \leq - 2, x \geq 2$

$(- \infty, - 2] \cup [2, \infty)$

$x \leq - 2, x \geq 2$

$(- \infty, - 2] \cup [2, \infty)$

Step 1.4

Evaluate

$- \sqrt{4 - x^{2}}$ 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

$x$ .

$- \sqrt{4 - {(0)}^{2}}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Raising

$0$ to any positive power yields

$0$ .

$- \sqrt{4 - 0}$

Step 1.4.1.2.2

Multiply

$- 1$ by

$0$ .

$- \sqrt{4 + 0}$

Step 1.4.1.2.3

Add

$4$ and

$0$ .

$- \sqrt{4}$

Step 1.4.1.2.4

Rewrite

$4$ as

$2^{2}$ .

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

Step 1.4.1.2.5

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

$- 1 \cdot 2$

Step 1.4.1.2.6

Multiply

$- 1$ by

$2$ .

$- 2$

Step 1.4.2

Evaluate at

$x = - 2$ .

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Step 1.4.2.1

Substitute

$- 2$ for

$x$ .

$- \sqrt{4 - {(- 2)}^{2}}$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Raise

$- 2$ to the power of

$2$ .

$- \sqrt{4 - 1 \cdot 4}$

Step 1.4.2.2.2

Multiply

$- 1$ by

$4$ .

$- \sqrt{4 - 4}$

Step 1.4.2.2.3

Subtract

$4$ from

$4$ .

$- \sqrt{0}$

Step 1.4.2.2.4

Rewrite

$0$ as

$0^{2}$ .

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

Step 1.4.2.2.5

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

$- 0$

Step 1.4.2.2.6

Multiply

$- 1$ by

$0$ .

$0$

Step 1.4.3

Evaluate at

$x = 2$ .

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Step 1.4.3.1

Substitute

$2$ for

$x$ .

$- \sqrt{4 - {(2)}^{2}}$

Step 1.4.3.2

Simplify.

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Step 1.4.3.2.1

Raise

$2$ to the power of

$2$ .

$- \sqrt{4 - 1 \cdot 4}$

Step 1.4.3.2.2

Multiply

$- 1$ by

$4$ .

$- \sqrt{4 - 4}$

Step 1.4.3.2.3

Subtract

$4$ from

$4$ .

$- \sqrt{0}$

Step 1.4.3.2.4

Rewrite

$0$ as

$0^{2}$ .

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

Step 1.4.3.2.5

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

$- 0$

Step 1.4.3.2.6

Multiply

$- 1$ by

$0$ .

$0$

Step 1.4.4

List all of the points.

$(0, - 2), (- 2, 0), (2, 0)$

Step 2

Exclude the points that are not on the interval.

$(0, - 2), (- 2, 0)$

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at

$x = - 2$ .

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Step 3.1.1

Substitute

$- 2$ for

$x$ .

$- \sqrt{4 - {(- 2)}^{2}}$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Raise

$- 2$ to the power of

$2$ .

$- \sqrt{4 - 1 \cdot 4}$

Step 3.1.2.2

Multiply

$- 1$ by

$4$ .

$- \sqrt{4 - 4}$

Step 3.1.2.3

Subtract

$4$ from

$4$ .

$- \sqrt{0}$

Step 3.1.2.4

Rewrite

$0$ as

$0^{2}$ .

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

Step 3.1.2.5

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

$- 0$

Step 3.1.2.6

Multiply

$- 1$ by

$0$ .

$0$

Step 3.2

Evaluate at

$x = 1$ .

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Step 3.2.1

Substitute

$1$ for

$x$ .

$- \sqrt{4 - {(1)}^{2}}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

One to any power is one.

$- \sqrt{4 - 1 \cdot 1}$

Step 3.2.2.2

Multiply

$- 1$ by

$1$ .

$- \sqrt{4 - 1}$

Step 3.2.2.3

Subtract

$1$ from

$4$ .

$- \sqrt{3}$

Step 3.3

List all of the points.

$(- 2, 0), (1, - \sqrt{3})$

Step 4

Compare the

$g (x)$ 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

$g (x)$ value and the minimum will occur at the lowest

$g (x)$ value.

Absolute Maximum:

$(- 2, 0)$

Absolute Minimum:

$(0, - 2)$

Step 5