Calculus Examples

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

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

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

Step 1.1.1.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(5 - x^{2})}^{\frac{1}{2}}$ with respect to $x$ is $- \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}]$ .

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

Step 1.1.1.2

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

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Step 1.1.1.2.1

To apply the Chain Rule, set $u$ as $5 - x^{2}$ .

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

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

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

Step 1.1.1.2.3

Replace all occurrences of $u$ with $5 - x^{2}$ .

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

Step 1.1.1.3

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

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

Step 1.1.1.4

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

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

Step 1.1.1.5

Combine the numerators over the common denominator.

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

Step 1.1.1.6

Simplify the numerator.

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Step 1.1.1.6.1

Multiply $- 1$ by $2$ .

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

Step 1.1.1.6.2

Subtract $2$ from $1$ .

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

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.

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

Step 1.1.1.7.2

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

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

Step 1.1.1.7.3

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

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

Step 1.1.1.8

By the Sum Rule, the derivative of $5 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- x^{2}]$ .

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

Step 1.1.1.9

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

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

Step 1.1.1.10

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

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

Step 1.1.1.11

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 1.1.1.12

Multiply.

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Step 1.1.1.12.1

Multiply $- 1$ by $- 1$ .

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

Step 1.1.1.12.2

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

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

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 $n = 2$ .

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

Step 1.1.1.14

Simplify terms.

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Step 1.1.1.14.1

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

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

Step 1.1.1.14.2

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

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

Step 1.1.1.14.3

Cancel the common factor.

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

Step 1.1.1.14.4

Rewrite the expression.

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

Step 1.1.1.14.5

Reorder terms.

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

Set the first derivative equal to $0$ .

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

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.

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

Step 1.3.1.2

Anything raised to $1$ is the base itself.

$\frac{x}{\sqrt{- x^{2} + 5}}$

Step 1.3.2

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

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

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.

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

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

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

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

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

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Step 1.3.3.2.2.1.1

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

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Step 1.3.3.2.2.1.1.1

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

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

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.

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

Step 1.3.3.2.2.1.1.2.2

Rewrite the expression.

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

Step 1.3.3.2.2.1.2

Simplify.

$- x^{2} + 5 = 0^{2}$

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 $0$ .

$- x^{2} + 5 = 0$

Step 1.3.3.3

Solve for $x$ .

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Step 1.3.3.3.1

Subtract $5$ from both sides of the equation.

$- x^{2} = - 5$

Step 1.3.3.3.2

Divide each term in $- x^{2} = - 5$ by $- 1$ and simplify.

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Step 1.3.3.3.2.1

Divide each term in $- x^{2} = - 5$ by $- 1$ .

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

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.

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

Step 1.3.3.3.2.2.2

Divide $x^{2}$ by $1$ .

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

Step 1.3.3.3.2.3

Simplify the right side.

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Step 1.3.3.3.2.3.1

Divide $- 5$ by $- 1$ .

$x^{2} = 5$

Step 1.3.3.3.3

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

$x = \pm \sqrt{5}$

Step 1.3.3.3.4

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

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Step 1.3.3.3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{5}$

Step 1.3.3.3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{5}$

Step 1.3.3.3.4.3

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

$x = \sqrt{5}, - \sqrt{5}$

Step 1.3.4

Set the radicand in $\sqrt{- x^{2} + 5}$ less than $0$ to find where the expression is undefined.

$- x^{2} + 5 < 0$

Step 1.3.5

Solve for $x$ .

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Step 1.3.5.1

Subtract $5$ from both sides of the inequality.

$- x^{2} < - 5$

Step 1.3.5.2

Divide each term in $- x^{2} < - 5$ by $- 1$ and simplify.

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Step 1.3.5.2.1

Divide each term in $- x^{2} < - 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} > \frac{- 5}{- 1}$

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.

$\frac{x^{2}}{1} > \frac{- 5}{- 1}$

Step 1.3.5.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} > \frac{- 5}{- 1}$

Step 1.3.5.2.3

Simplify the right side.

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Step 1.3.5.2.3.1

Divide $- 5$ by $- 1$ .

$x^{2} > 5$

Step 1.3.5.3

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

$\sqrt{x^{2}} > \sqrt{5}$

Step 1.3.5.4

Simplify the left side.

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Step 1.3.5.4.1

Pull terms out from under the radical.

$| x | > \sqrt{5}$

Step 1.3.5.5

Write $| x | > \sqrt{5}$ 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.

$x > \sqrt{5}$

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 $- 1$ .

$- x > \sqrt{5}$

Step 1.3.5.5.5

Write as a piecewise.

${\begin{cases} x > \sqrt{5} & x \geq 0 \\ - x > \sqrt{5} & x < 0 \end{cases}$

Step 1.3.5.6

Find the intersection of $x > \sqrt{5}$ and $x \geq 0$ .

$x > \sqrt{5}$

Step 1.3.5.7

Divide each term in $- x > \sqrt{5}$ by $- 1$ and simplify.

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Step 1.3.5.7.1

Divide each term in $- x > \sqrt{5}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

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.

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

Step 1.3.5.7.2.2

Divide $x$ by $1$ .

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

Step 1.3.5.7.3

Simplify the right side.

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Step 1.3.5.7.3.1

Move the negative one from the denominator of $\frac{\sqrt{5}}{- 1}$ .

$x < - 1 \cdot \sqrt{5}$

Step 1.3.5.7.3.2

Rewrite $- 1 \cdot \sqrt{5}$ as $- \sqrt{5}$ .

$x < - \sqrt{5}$

Step 1.3.5.8

Find the union of the solutions.

$x < - \sqrt{5}$ or $x > \sqrt{5}$

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 $0$ .

$x \leq - \sqrt{5}, x \geq \sqrt{5}$

$(- \infty, - \sqrt{5}] \cup [\sqrt{5}, \infty)$

$x \leq - \sqrt{5}, x \geq \sqrt{5}$

$(- \infty, - \sqrt{5}] \cup [\sqrt{5}, \infty)$

Step 1.4

Evaluate $- \sqrt{5 - x^{2}}$ 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 $x$ .

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

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

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

$- \sqrt{5 - 0}$

Step 1.4.1.2.2

Multiply $- 1$ by $0$ .

$- \sqrt{5 + 0}$

Step 1.4.1.2.3

Add $5$ and $0$ .

$- \sqrt{5}$

Step 1.4.2

Evaluate at $x = - \sqrt{5}$ .

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Step 1.4.2.1

Substitute $- \sqrt{5}$ for $x$ .

$- \sqrt{5 - {(- \sqrt{5})}^{2}}$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Apply the product rule to $- \sqrt{5}$ .

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

Step 1.4.2.2.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 1.4.2.2.2.1

Move ${(- 1)}^{2}$ .

$- \sqrt{5 + {(- 1)}^{2} \cdot - 1 {\sqrt{5}}^{2}}$

Step 1.4.2.2.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 1.4.2.2.2.2.1

Raise $- 1$ to the power of $1$ .

$- \sqrt{5 + {(- 1)}^{2} \cdot {(- 1)}^{1} {\sqrt{5}}^{2}}$

Step 1.4.2.2.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.4.2.2.2.3

Add $2$ and $1$ .

$- \sqrt{5 + {(- 1)}^{3} {\sqrt{5}}^{2}}$

Step 1.4.2.2.3

Raise $- 1$ to the power of $3$ .

$- \sqrt{5 - {\sqrt{5}}^{2}}$

Step 1.4.2.2.4

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 1.4.2.2.4.1

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

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

Step 1.4.2.2.4.2

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

$- \sqrt{5 - 5^{\frac{1}{2} \cdot 2}}$

Step 1.4.2.2.4.3

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

$- \sqrt{5 - 5^{\frac{2}{2}}}$

Step 1.4.2.2.4.4

Cancel the common factor of $2$ .

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Step 1.4.2.2.4.4.1

Cancel the common factor.

$- \sqrt{5 - 5^{\frac{2}{2}}}$

Step 1.4.2.2.4.4.2

Rewrite the expression.

$- \sqrt{5 - 5^{1}}$

Step 1.4.2.2.4.5

Evaluate the exponent.

$- \sqrt{5 - 1 \cdot 5}$

Step 1.4.2.2.5

Simplify the expression.

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Step 1.4.2.2.5.1

Multiply $- 1$ by $5$ .

$- \sqrt{5 - 5}$

Step 1.4.2.2.5.2

Subtract $5$ from $5$ .

$- \sqrt{0}$

Step 1.4.2.2.5.3

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

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

Step 1.4.2.2.5.4

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

$- 0$

Step 1.4.2.2.5.5

Multiply $- 1$ by $0$ .

$0$

Step 1.4.3

Evaluate at $x = \sqrt{5}$ .

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Step 1.4.3.1

Substitute $\sqrt{5}$ for $x$ .

$- \sqrt{5 - {(\sqrt{5})}^{2}}$

Step 1.4.3.2

Simplify.

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Step 1.4.3.2.1

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 1.4.3.2.1.1

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

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

Step 1.4.3.2.1.2

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

$- \sqrt{5 - 5^{\frac{1}{2} \cdot 2}}$

Step 1.4.3.2.1.3

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

$- \sqrt{5 - 5^{\frac{2}{2}}}$

Step 1.4.3.2.1.4

Cancel the common factor of $2$ .

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Step 1.4.3.2.1.4.1

Cancel the common factor.

$- \sqrt{5 - 5^{\frac{2}{2}}}$

Step 1.4.3.2.1.4.2

Rewrite the expression.

$- \sqrt{5 - 5^{1}}$

Step 1.4.3.2.1.5

Evaluate the exponent.

$- \sqrt{5 - 1 \cdot 5}$

Step 1.4.3.2.2

Simplify the expression.

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Step 1.4.3.2.2.1

Multiply $- 1$ by $5$ .

$- \sqrt{5 - 5}$

Step 1.4.3.2.2.2

Subtract $5$ from $5$ .

$- \sqrt{0}$

Step 1.4.3.2.2.3

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

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

Step 1.4.3.2.2.4

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

$- 0$

Step 1.4.3.2.2.5

Multiply $- 1$ by $0$ .

$0$

Step 1.4.4

List all of the points.

$(0, - \sqrt{5}), (- \sqrt{5}, 0), (\sqrt{5}, 0)$

Step 2

Exclude the points that are not on the interval.

$(0, - \sqrt{5}), (- \sqrt{5}, 0)$

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = - \sqrt{5}$ .

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Step 3.1.1

Substitute $- \sqrt{5}$ for $x$ .

$- \sqrt{5 - {(- \sqrt{5})}^{2}}$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Apply the product rule to $- \sqrt{5}$ .

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

Step 3.1.2.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 3.1.2.2.1

Move ${(- 1)}^{2}$ .

$- \sqrt{5 + {(- 1)}^{2} \cdot - 1 {\sqrt{5}}^{2}}$

Step 3.1.2.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 3.1.2.2.2.1

Raise $- 1$ to the power of $1$ .

$- \sqrt{5 + {(- 1)}^{2} \cdot {(- 1)}^{1} {\sqrt{5}}^{2}}$

Step 3.1.2.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.1.2.2.3

Add $2$ and $1$ .

$- \sqrt{5 + {(- 1)}^{3} {\sqrt{5}}^{2}}$

Step 3.1.2.3

Raise $- 1$ to the power of $3$ .

$- \sqrt{5 - {\sqrt{5}}^{2}}$

Step 3.1.2.4

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.1.2.4.1

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

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

Step 3.1.2.4.2

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

$- \sqrt{5 - 5^{\frac{1}{2} \cdot 2}}$

Step 3.1.2.4.3

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

$- \sqrt{5 - 5^{\frac{2}{2}}}$

Step 3.1.2.4.4

Cancel the common factor of $2$ .

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Step 3.1.2.4.4.1

Cancel the common factor.

$- \sqrt{5 - 5^{\frac{2}{2}}}$

Step 3.1.2.4.4.2

Rewrite the expression.

$- \sqrt{5 - 5^{1}}$

Step 3.1.2.4.5

Evaluate the exponent.

$- \sqrt{5 - 1 \cdot 5}$

Step 3.1.2.5

Simplify the expression.

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Step 3.1.2.5.1

Multiply $- 1$ by $5$ .

$- \sqrt{5 - 5}$

Step 3.1.2.5.2

Subtract $5$ from $5$ .

$- \sqrt{0}$

Step 3.1.2.5.3

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

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

Step 3.1.2.5.4

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

$- 0$

Step 3.1.2.5.5

Multiply $- 1$ by $0$ .

$0$

Step 3.2

Evaluate at $x = 0$ .

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Step 3.2.1

Substitute $0$ for $x$ .

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

Step 3.2.2

Simplify.

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Step 3.2.2.1

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

$- \sqrt{5 - 0}$

Step 3.2.2.2

Multiply $- 1$ by $0$ .

$- \sqrt{5 + 0}$

Step 3.2.2.3

Add $5$ and $0$ .

$- \sqrt{5}$

Step 3.3

List all of the points.

$(- \sqrt{5}, 0), (0, - \sqrt{5})$

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: $(- \sqrt{5}, 0)$

Absolute Minimum: $(0, - \sqrt{5})$

Step 5