Calculus Examples

$g (x) = \sqrt{1 - x^{2}}$ , $- 1 \leq x \leq 0$

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

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

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

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 $g (x) = 1 - x^{2}$ .

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Step 1.1.1.2.1

To apply the Chain Rule, set $u$ as $1 - x^{2}$ .

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

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

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

Step 1.1.1.2.3

Replace all occurrences of $u$ with $1 - x^{2}$ .

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

Step 1.1.1.3

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

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

Step 1.1.1.4

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

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

Step 1.1.1.5

Combine the numerators over the common denominator.

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

Step 1.1.1.6

Simplify the numerator.

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Step 1.1.1.6.1

Multiply $- 1$ by $2$ .

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

Step 1.1.1.6.2

Subtract $2$ from $1$ .

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

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.

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

Step 1.1.1.7.2

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

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

Step 1.1.1.7.3

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

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

Step 1.1.1.8

By the Sum Rule, the derivative of $1 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}]$ .

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

Step 1.1.1.9

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

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

Step 1.1.1.10

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

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

Step 1.1.1.11

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

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

Step 1.1.1.12

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

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

Step 1.1.1.13

Simplify terms.

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Step 1.1.1.13.1

Multiply $2$ by $- 1$ .

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

Step 1.1.1.13.2

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

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

Step 1.1.1.13.3

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

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

Step 1.1.1.13.4

Factor $2$ out of $- 2 x$ .

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

Step 1.1.1.14

Cancel the common factors.

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Step 1.1.1.14.1

Factor $2$ out of $2 {(1 - x^{2})}^{\frac{1}{2}}$ .

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

Step 1.1.1.14.2

Cancel the common factor.

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

Step 1.1.1.14.3

Rewrite the expression.

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

Step 1.1.1.15

Move the negative in front of the fraction.

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

Set the first derivative equal to $0$ .

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

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.

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

Step 1.3.1.2

Anything raised to $1$ is the base itself.

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

Step 1.3.2

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

$\sqrt{1 - x^{2}} = 0$

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.

${\sqrt{1 - x^{2}}}^{2} = 0^{2}$

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

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

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

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

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Step 1.3.3.2.2.1.1

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

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Step 1.3.3.2.2.1.1.1

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

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

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.

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

Step 1.3.3.2.2.1.1.2.2

Rewrite the expression.

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

Step 1.3.3.2.2.1.2

Simplify.

$1 - x^{2} = 0^{2}$

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 $0$ .

$1 - x^{2} = 0$

Step 1.3.3.3

Solve for $x$ .

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Step 1.3.3.3.1

Subtract $1$ from both sides of the equation.

$- x^{2} = - 1$

Step 1.3.3.3.2

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

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Step 1.3.3.3.2.1

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

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

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.

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

Step 1.3.3.3.2.2.2

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

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

Step 1.3.3.3.2.3

Simplify the right side.

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Step 1.3.3.3.2.3.1

Divide $- 1$ by $- 1$ .

$x^{2} = 1$

Step 1.3.3.3.3

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

$x = \pm \sqrt{1}$

Step 1.3.3.3.4

Any root of $1$ is $1$ .

$x = \pm 1$

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.

$x = 1$

Step 1.3.3.3.5.2

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

$x = - 1$

Step 1.3.3.3.5.3

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

$x = 1, - 1$

Step 1.3.4

Set the radicand in $\sqrt{1 - x^{2}}$ less than $0$ to find where the expression is undefined.

$1 - x^{2} < 0$

Step 1.3.5

Solve for $x$ .

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Step 1.3.5.1

Subtract $1$ from both sides of the inequality.

$- x^{2} < - 1$

Step 1.3.5.2

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

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Step 1.3.5.2.1

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

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

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.

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

Step 1.3.5.2.2.2

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

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

Step 1.3.5.2.3

Simplify the right side.

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Step 1.3.5.2.3.1

Divide $- 1$ by $- 1$ .

$x^{2} > 1$

Step 1.3.5.3

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

$\sqrt{x^{2}} > \sqrt{1}$

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.

$| x | > \sqrt{1}$

Step 1.3.5.4.2

Simplify the right side.

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Step 1.3.5.4.2.1

Any root of $1$ is $1$ .

$| x | > 1$

Step 1.3.5.5

Write $| x | > 1$ 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.

$x > 1$

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 $- 1$ .

$- x > 1$

Step 1.3.5.5.5

Write as a piecewise.

${\begin{cases} x > 1 & x \geq 0 \\ - x > 1 & x < 0 \end{cases}$

Step 1.3.5.6

Find the intersection of $x > 1$ and $x \geq 0$ .

$x > 1$

Step 1.3.5.7

Divide each term in $- x > 1$ by $- 1$ and simplify.

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Step 1.3.5.7.1

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

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

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.

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

Step 1.3.5.7.2.2

Divide $x$ by $1$ .

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

Step 1.3.5.7.3

Simplify the right side.

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Step 1.3.5.7.3.1

Divide $1$ by $- 1$ .

$x < - 1$

Step 1.3.5.8

Find the union of the solutions.

$x < - 1$ or $x > 1$

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 $0$ .

$x \leq - 1, x \geq 1$

$(- \infty, - 1] \cup [1, \infty)$

$x \leq - 1, x \geq 1$

$(- \infty, - 1] \cup [1, \infty)$

Step 1.4

Evaluate $\sqrt{1 - x^{2}}$ 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 $x$ .

$\sqrt{1 - {(0)}^{2}}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

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

$\sqrt{1 - 0}$

Step 1.4.1.2.2

Multiply $- 1$ by $0$ .

$\sqrt{1 + 0}$

Step 1.4.1.2.3

Add $1$ and $0$ .

$\sqrt{1}$

Step 1.4.1.2.4

Any root of $1$ is $1$ .

$1$

Step 1.4.2

Evaluate at $x = - 1$ .

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Step 1.4.2.1

Substitute $- 1$ for $x$ .

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

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

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

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Step 1.4.2.2.1.1

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

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Step 1.4.2.2.1.1.1

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

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

Step 1.4.2.2.1.1.2

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

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

Step 1.4.2.2.1.2

Add $1$ and $2$ .

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

Step 1.4.2.2.2

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

$\sqrt{1 - 1}$

Step 1.4.2.2.3

Subtract $1$ from $1$ .

$\sqrt{0}$

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 1.4.3

Evaluate at $x = 1$ .

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Step 1.4.3.1

Substitute $1$ for $x$ .

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

Step 1.4.3.2

Simplify.

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Step 1.4.3.2.1

One to any power is one.

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

Step 1.4.3.2.2

Multiply $- 1$ by $1$ .

$\sqrt{1 - 1}$

Step 1.4.3.2.3

Subtract $1$ from $1$ .

$\sqrt{0}$

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 1.4.4

List all of the points.

$(0, 1), (- 1, 0), (1, 0)$

Step 2

Exclude the points that are not on the interval.

$(0, 1), (- 1, 0)$

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = - 1$ .

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Step 3.1.1

Substitute $- 1$ for $x$ .

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

Step 3.1.2

Simplify.

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Step 3.1.2.1

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

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Step 3.1.2.1.1

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

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Step 3.1.2.1.1.1

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

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

Step 3.1.2.1.1.2

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

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

Step 3.1.2.1.2

Add $1$ and $2$ .

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

Step 3.1.2.2

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

$\sqrt{1 - 1}$

Step 3.1.2.3

Subtract $1$ from $1$ .

$\sqrt{0}$

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 3.2

Evaluate at $x = 0$ .

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Step 3.2.1

Substitute $0$ for $x$ .

$\sqrt{1 - {(0)}^{2}}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

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

$\sqrt{1 - 0}$

Step 3.2.2.2

Multiply $- 1$ by $0$ .

$\sqrt{1 + 0}$

Step 3.2.2.3

Add $1$ and $0$ .

$\sqrt{1}$

Step 3.2.2.4

Any root of $1$ is $1$ .

$1$

Step 3.3

List all of the points.

$(- 1, 0), (0, 1)$

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: $(0, 1)$

Absolute Minimum: $(- 1, 0)$

Step 5