Calculus Examples

$f (x) = \sqrt{20 - x - x^{2}}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{20 - x - x^{2}}$ as ${(20 - x - x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(20 - x - x^{2})}^{\frac{1}{2}}]$

Step 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 $g (x) = 20 - x - x^{2}$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $20 - x - x^{2}$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [20 - x - x^{2}]$

Step 1.1.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [20 - x - x^{2}]$

Step 1.1.2.3

Replace all occurrences of $u$ with $20 - x - x^{2}$ .

$\frac{1}{2} {(20 - x - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [20 - x - x^{2}]$

Step 1.1.3

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

$\frac{1}{2} {(20 - x - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [20 - x - x^{2}]$

Step 1.1.4

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

$\frac{1}{2} {(20 - x - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [20 - x - x^{2}]$

Step 1.1.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(20 - x - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [20 - x - x^{2}]$

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(20 - x - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [20 - x - x^{2}]$

Step 1.1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(20 - x - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} [20 - x - x^{2}]$

Step 1.1.7

Combine fractions.

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Step 1.1.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(20 - x - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} [20 - x - x^{2}]$

Step 1.1.7.2

Combine $\frac{1}{2}$ and ${(20 - x - x^{2})}^{- \frac{1}{2}}$ .

$\frac{{(20 - x - x^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [20 - x - x^{2}]$

Step 1.1.7.3

Move ${(20 - x - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(20 - x - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [20 - x - x^{2}]$

Step 1.1.8

By the Sum Rule, the derivative of $20 - x - x^{2}$ with respect to $x$ is $\frac{d}{d x} [20] + \frac{d}{d x} [- x] + \frac{d}{d x} [- x^{2}]$ .

$\frac{1}{2 {(20 - x - x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [20] + \frac{d}{d x} [- x] + \frac{d}{d x} [- x^{2}])$

Step 1.1.9

Since $20$ is constant with respect to $x$ , the derivative of $20$ with respect to $x$ is $0$ .

$\frac{1}{2 {(20 - x - x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x] + \frac{d}{d x} [- x^{2}])$

Step 1.1.10

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

$\frac{1}{2 {(20 - x - x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [- x] + \frac{d}{d x} [- x^{2}])$

Step 1.1.11

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

$\frac{1}{2 {(20 - x - x^{2})}^{\frac{1}{2}}} (- \frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}])$

Step 1.1.12

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

$\frac{1}{2 {(20 - x - x^{2})}^{\frac{1}{2}}} (- 1 \cdot 1 + \frac{d}{d x} [- x^{2}])$

Step 1.1.13

Multiply $- 1$ by $1$ .

$\frac{1}{2 {(20 - x - x^{2})}^{\frac{1}{2}}} (- 1 + \frac{d}{d x} [- x^{2}])$

Step 1.1.14

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

$\frac{1}{2 {(20 - x - x^{2})}^{\frac{1}{2}}} (- 1 - \frac{d}{d x} [x^{2}])$

Step 1.1.15

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

$\frac{1}{2 {(20 - x - x^{2})}^{\frac{1}{2}}} (- 1 - (2 x))$

Step 1.1.16

Multiply $2$ by $- 1$ .

$\frac{1}{2 {(20 - x - x^{2})}^{\frac{1}{2}}} (- 1 - 2 x)$

Step 1.1.17

Simplify.

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Step 1.1.17.1

Reorder the factors of $\frac{1}{2 {(20 - x - x^{2})}^{\frac{1}{2}}} (- 1 - 2 x)$ .

$(- 1 - 2 x) \frac{1}{2 {(20 - x - x^{2})}^{\frac{1}{2}}}$

Step 1.1.17.2

Multiply $- 1 - 2 x$ by $\frac{1}{2 {(20 - x - x^{2})}^{\frac{1}{2}}}$ .

$\frac{- 1 - 2 x}{2 {(20 - x - x^{2})}^{\frac{1}{2}}}$

Step 1.1.17.3

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- 1 (1) - 2 x}{2 {(20 - x - x^{2})}^{\frac{1}{2}}}$

Step 1.1.17.4

Factor $- 1$ out of $- 2 x$ .

$\frac{- 1 (1) - (2 x)}{2 {(20 - x - x^{2})}^{\frac{1}{2}}}$

Step 1.1.17.5

Factor $- 1$ out of $- 1 (1) - (2 x)$ .

$\frac{- 1 (1 + 2 x)}{2 {(20 - x - x^{2})}^{\frac{1}{2}}}$

Step 1.1.17.6

Move the negative in front of the fraction.

$f' (x) = - \frac{1 + 2 x}{2 {(20 - x - x^{2})}^{\frac{1}{2}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{1 + 2 x}{2 {(20 - x - x^{2})}^{\frac{1}{2}}}$ .

$- \frac{1 + 2 x}{2 {(20 - x - x^{2})}^{\frac{1}{2}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{1 + 2 x}{2 {(20 - x - x^{2})}^{\frac{1}{2}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{1 + 2 x}{2 {(20 - x - x^{2})}^{\frac{1}{2}}} = 0$

Step 2.2

Set the numerator equal to zero.

$1 + 2 x = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 2.3.2

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

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Step 2.3.2.1

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

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

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

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

Step 2.3.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Move the negative in front of the fraction.

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

Step 3

The values which make the derivative equal to $0$ are $- \frac{1}{2}$ .

$- \frac{1}{2}$

Step 4

Find where the derivative is undefined.

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Step 4.1

Convert expressions with fractional exponents to radicals.

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Step 4.1.1

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

$- \frac{1 + 2 x}{2 \sqrt{{(20 - x - x^{2})}^{1}}}$

Step 4.1.2

Anything raised to $1$ is the base itself.

$- \frac{1 + 2 x}{2 \sqrt{20 - x - x^{2}}}$

Step 4.2

Set the denominator in $\frac{1 + 2 x}{2 \sqrt{20 - x - x^{2}}}$ equal to $0$ to find where the expression is undefined.

$2 \sqrt{20 - x - x^{2}} = 0$

Step 4.3

Solve for $x$ .

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Step 4.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${(2 \sqrt{20 - x - x^{2}})}^{2} = 0^{2}$

Step 4.3.2

Simplify each side of the equation.

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Step 4.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{20 - x - x^{2}}$ as ${(20 - x - x^{2})}^{\frac{1}{2}}$ .

${(2 {(20 - x - x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Simplify ${(2 {(20 - x - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.2.2.1.1

Apply the product rule to $2 {(20 - x - x^{2})}^{\frac{1}{2}}$ .

$2^{2} {({(20 - x - x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.3.2.2.1.2

Raise $2$ to the power of $2$ .

$4 {({(20 - x - x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.3.2.2.1.3

Multiply the exponents in ${({(20 - x - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.2.2.1.3.1

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

$4 {(20 - x - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 4.3.2.2.1.3.2.1

Cancel the common factor.

$4 {(20 - x - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.3.2.2.1.3.2.2

Rewrite the expression.

$4 {(20 - x - x^{2})}^{1} = 0^{2}$

Step 4.3.2.2.1.4

Simplify.

$4 (20 - x - x^{2}) = 0^{2}$

Step 4.3.2.2.1.5

Apply the distributive property.

$4 \cdot 20 + 4 (- x) + 4 (- x^{2}) = 0^{2}$

Step 4.3.2.2.1.6

Simplify.

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Step 4.3.2.2.1.6.1

Multiply $4$ by $20$ .

$80 + 4 (- x) + 4 (- x^{2}) = 0^{2}$

Step 4.3.2.2.1.6.2

Multiply $- 1$ by $4$ .

$80 - 4 x + 4 (- x^{2}) = 0^{2}$

Step 4.3.2.2.1.6.3

Multiply $- 1$ by $4$ .

$80 - 4 x - 4 x^{2} = 0^{2}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

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

$80 - 4 x - 4 x^{2} = 0$

Step 4.3.3

Solve for $x$ .

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Step 4.3.3.1

Factor the left side of the equation.

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Step 4.3.3.1.1

Factor $- 4$ out of $80 - 4 x - 4 x^{2}$ .

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Step 4.3.3.1.1.1

Reorder the expression.

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Step 4.3.3.1.1.1.1

Move $80$ .

$- 4 x - 4 x^{2} + 80 = 0$

Step 4.3.3.1.1.1.2

Reorder $- 4 x$ and $- 4 x^{2}$ .

$- 4 x^{2} - 4 x + 80 = 0$

Step 4.3.3.1.1.2

Factor $- 4$ out of $- 4 x^{2}$ .

$- 4 x^{2} - 4 x + 80 = 0$

Step 4.3.3.1.1.3

Factor $- 4$ out of $- 4 x$ .

$- 4 x^{2} - 4 x + 80 = 0$

Step 4.3.3.1.1.4

Factor $- 4$ out of $80$ .

$- 4 x^{2} - 4 x - 4 \cdot - 20 = 0$

Step 4.3.3.1.1.5

Factor $- 4$ out of $- 4 (x^{2}) - 4 (x)$ .

$- 4 (x^{2} + x) - 4 \cdot - 20 = 0$

Step 4.3.3.1.1.6

Factor $- 4$ out of $- 4 (x^{2} + x) - 4 (- 20)$ .

$- 4 (x^{2} + x - 20) = 0$

Step 4.3.3.1.2

Factor.

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Step 4.3.3.1.2.1

Factor $x^{2} + x - 20$ using the AC method.

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Step 4.3.3.1.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 20$ and whose sum is $1$ .

$- 4, 5$

Step 4.3.3.1.2.1.2

Write the factored form using these integers.

$- 4 ((x - 4) (x + 5)) = 0$

Step 4.3.3.1.2.2

Remove unnecessary parentheses.

$- 4 (x - 4) (x + 5) = 0$

Step 4.3.3.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 4 = 0$

$x + 5 = 0$

Step 4.3.3.3

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 4.3.3.3.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 4.3.3.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 4.3.3.4

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 4.3.3.4.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 4.3.3.4.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 4.3.3.5

The final solution is all the values that make $- 4 (x - 4) (x + 5) = 0$ true.

$x = 4, - 5$

Step 4.4

Set the radicand in $\sqrt{20 - x - x^{2}}$ less than $0$ to find where the expression is undefined.

$20 - x - x^{2} < 0$

Step 4.5

Solve for $x$ .

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Step 4.5.1

Convert the inequality to an equation.

$20 - x - x^{2} = 0$

Step 4.5.2

Factor the left side of the equation.

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Step 4.5.2.1

Factor $- 1$ out of $20 - x - x^{2}$ .

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Step 4.5.2.1.1

Reorder the expression.

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Step 4.5.2.1.1.1

Move $20$ .

$- x - x^{2} + 20 = 0$

Step 4.5.2.1.1.2

Reorder $- x$ and $- x^{2}$ .

$- x^{2} - x + 20 = 0$

Step 4.5.2.1.2

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) - x + 20 = 0$

Step 4.5.2.1.3

Factor $- 1$ out of $- x$ .

$- (x^{2}) - (x) + 20 = 0$

Step 4.5.2.1.4

Rewrite $20$ as $- 1 (- 20)$ .

$- (x^{2}) - (x) - 1 \cdot - 20 = 0$

Step 4.5.2.1.5

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

$- (x^{2} + x) - 1 \cdot - 20 = 0$

Step 4.5.2.1.6

Factor $- 1$ out of $- (x^{2} + x) - 1 (- 20)$ .

$- (x^{2} + x - 20) = 0$

Step 4.5.2.2

Factor.

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Step 4.5.2.2.1

Factor $x^{2} + x - 20$ using the AC method.

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Step 4.5.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 20$ and whose sum is $1$ .

$- 4, 5$

Step 4.5.2.2.1.2

Write the factored form using these integers.

$- ((x - 4) (x + 5)) = 0$

Step 4.5.2.2.2

Remove unnecessary parentheses.

$- (x - 4) (x + 5) = 0$

Step 4.5.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 4 = 0$

$x + 5 = 0$

Step 4.5.4

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 4.5.4.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 4.5.4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 4.5.5

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 4.5.5.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 4.5.5.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 4.5.6

The final solution is all the values that make $- (x - 4) (x + 5) = 0$ true.

$x = 4, - 5$

Step 4.5.7

Use each root to create test intervals.

$x < - 5$

$- 5 < x < 4$

$x > 4$

Step 4.5.8

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 4.5.8.1

Test a value on the interval $x < - 5$ to see if it makes the inequality true.

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Step 4.5.8.1.1

Choose a value on the interval $x < - 5$ and see if this value makes the original inequality true.

$x = - 8$

Step 4.5.8.1.2

Replace $x$ with $- 8$ in the original inequality.

$20 - (- 8) - {(- 8)}^{2} < 0$

Step 4.5.8.1.3

The left side $- 36$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 4.5.8.2

Test a value on the interval $- 5 < x < 4$ to see if it makes the inequality true.

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Step 4.5.8.2.1

Choose a value on the interval $- 5 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

Step 4.5.8.2.2

Replace $x$ with $0$ in the original inequality.

$20 - (0) - {(0)}^{2} < 0$

Step 4.5.8.2.3

The left side $20$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 4.5.8.3

Test a value on the interval $x > 4$ to see if it makes the inequality true.

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Step 4.5.8.3.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 4.5.8.3.2

Replace $x$ with $6$ in the original inequality.

$20 - (6) - {(6)}^{2} < 0$

Step 4.5.8.3.3

The left side $- 22$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 4.5.8.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 5$ True

$- 5 < x < 4$ False

$x > 4$ True

$x < - 5$ True

$- 5 < x < 4$ False

$x > 4$ True

Step 4.5.9

The solution consists of all of the true intervals.

$x < - 5$ or $x > 4$

Step 4.6

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 $0$ .

$x \leq - 5, x \geq 4$

$(- \infty, - 5] \cup [4, \infty)$

$x \leq - 5, x \geq 4$

$(- \infty, - 5] \cup [4, \infty)$

Step 5

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, - 5) \cup (- 5, - \frac{1}{2}) \cup (- \frac{1}{2}, 4) \cup (4, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 5)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 6$ in the expression.

$f' (- 6) = - \frac{1 + 2 (- 6)}{2 {(20 - (- 6) - {(- 6)}^{2})}^{\frac{1}{2}}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Multiply $2$ by $- 6$ .

$f' (- 6) = - \frac{1 - 12}{2 {(20 - (- 6) - {(- 6)}^{2})}^{\frac{1}{2}}}$

Step 6.2.1.2

Subtract $12$ from $1$ .

$f' (- 6) = - \frac{- 11}{2 {(20 - (- 6) - {(- 6)}^{2})}^{\frac{1}{2}}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

Multiply $- 1$ by $- 6$ .

$f' (- 6) = - \frac{- 11}{2 {(20 + 6 - {(- 6)}^{2})}^{\frac{1}{2}}}$

Step 6.2.2.1.2

Raise $- 6$ to the power of $2$ .

$f' (- 6) = - \frac{- 11}{2 {(20 + 6 - 1 \cdot 36)}^{\frac{1}{2}}}$

Step 6.2.2.1.3

Multiply $- 1$ by $36$ .

$f' (- 6) = - \frac{- 11}{2 {(20 + 6 - 36)}^{\frac{1}{2}}}$

Step 6.2.2.2

Add $20$ and $6$ .

$f' (- 6) = - \frac{- 11}{2 {(26 - 36)}^{\frac{1}{2}}}$

Step 6.2.2.3

Subtract $36$ from $26$ .

$f' (- 6) = - \frac{- 11}{2 {(- 10)}^{\frac{1}{2}}}$

Step 6.2.3

Move the negative in front of the fraction.

$f' (- 6) = \frac{11}{2 {(- 10)}^{\frac{1}{2}}}$

Step 6.2.4

The final answer is $\frac{11}{2 {(- 10)}^{\frac{1}{2}}}$ .

$\frac{11}{2 {(- 10)}^{\frac{1}{2}}}$

Step 6.3

At $x = - 6$ the derivative is $\frac{11}{2 {(- 10)}^{\frac{1}{2}}}$ . Since this is negative, the function is decreasing on $(- \infty, - 5)$ .

Decreasing on $(- \infty, - 5)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(- 5, - \frac{1}{2})$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 2.75$ in the expression.

$f' (- 2.75) = - \frac{1 + 2 (- 2.75)}{2 {(20 - (- 2.75) - {(- 2.75)}^{2})}^{\frac{1}{2}}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Multiply $2$ by $- 2.75$ .

$f' (- 2.75) = - \frac{1 - 5.5}{2 {(20 - (- 2.75) - {(- 2.75)}^{2})}^{\frac{1}{2}}}$

Step 7.2.1.2

Subtract $5.5$ from $1$ .

$f' (- 2.75) = - \frac{- 4.5}{2 {(20 - (- 2.75) - {(- 2.75)}^{2})}^{\frac{1}{2}}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Simplify each term.

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Step 7.2.2.1.1

Multiply $- 1$ by $- 2.75$ .

$f' (- 2.75) = - \frac{- 4.5}{2 {(20 + 2.75 - {(- 2.75)}^{2})}^{\frac{1}{2}}}$

Step 7.2.2.1.2

Raise $- 2.75$ to the power of $2$ .

$f' (- 2.75) = - \frac{- 4.5}{2 {(20 + 2.75 - 1 \cdot 7.5625)}^{\frac{1}{2}}}$

Step 7.2.2.1.3

Multiply $- 1$ by $7.5625$ .

$f' (- 2.75) = - \frac{- 4.5}{2 {(20 + 2.75 - 7.5625)}^{\frac{1}{2}}}$

Step 7.2.2.2

Add $20$ and $2.75$ .

$f' (- 2.75) = - \frac{- 4.5}{2 {(22.75 - 7.5625)}^{\frac{1}{2}}}$

Step 7.2.2.3

Subtract $7.5625$ from $22.75$ .

$f' (- 2.75) = - \frac{- 4.5}{2 \cdot {15.1875}^{\frac{1}{2}}}$

Step 7.2.2.4

Rewrite $15.1875$ as ${3.89711431}^{2}$ .

$f' (- 2.75) = - \frac{- 4.5}{2 \cdot {({3.89711431}^{2})}^{\frac{1}{2}}}$

Step 7.2.2.5

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

$f' (- 2.75) = - \frac{- 4.5}{2 \cdot {3.89711431}^{2 (\frac{1}{2})}}$

Step 7.2.2.6

Cancel the common factor of $2$ .

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Step 7.2.2.6.1

Cancel the common factor.

$f' (- 2.75) = - \frac{- 4.5}{2 \cdot {3.89711431}^{2 (\frac{1}{2})}}$

Step 7.2.2.6.2

Rewrite the expression.

$f' (- 2.75) = - \frac{- 4.5}{2 \cdot 3.89711431}$

Step 7.2.2.7

Evaluate the exponent.

$f' (- 2.75) = - \frac{- 4.5}{2 \cdot 3.89711431}$

Step 7.2.3

Simplify the expression.

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Step 7.2.3.1

Multiply $2$ by $3.89711431$ .

$f' (- 2.75) = - \frac{- 4.5}{7.79422863}$

Step 7.2.3.2

Divide $- 4.5$ by $7.79422863$ .

$f' (- 2.75) = 0.57735026$

Step 7.2.3.3

Multiply $- 1$ by $- 0.57735026$ .

$f' (- 2.75) = 0.57735026$

Step 7.2.4

The final answer is $0.57735026$ .

$0.57735026$

Step 7.3

At $x = - 2.75$ the derivative is $0.57735026$ . Since this is positive, the function is increasing on $(- 5, - \frac{1}{2})$ .

Increasing on $(- 5, - \frac{1}{2})$ since $f' (x) > 0$

Step 8

Substitute a value from the interval $(- 0.5, 4)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $1.75$ in the expression.

$f' (1.75) = - \frac{1 + 2 (1.75)}{2 {(20 - (1.75) - {(1.75)}^{2})}^{\frac{1}{2}}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify the numerator.

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Step 8.2.1.1

Multiply $2$ by $1.75$ .

$f' (1.75) = - \frac{1 + 3.5}{2 {(20 - (1.75) - {1.75}^{2})}^{\frac{1}{2}}}$

Step 8.2.1.2

Add $1$ and $3.5$ .

$f' (1.75) = - \frac{4.5}{2 {(20 - (1.75) - {1.75}^{2})}^{\frac{1}{2}}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Simplify each term.

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Step 8.2.2.1.1

Multiply $- 1$ by $1.75$ .

$f' (1.75) = - \frac{4.5}{2 {(20 - 1.75 - {1.75}^{2})}^{\frac{1}{2}}}$

Step 8.2.2.1.2

Raise $1.75$ to the power of $2$ .

$f' (1.75) = - \frac{4.5}{2 {(20 - 1.75 - 1 \cdot 3.0625)}^{\frac{1}{2}}}$

Step 8.2.2.1.3

Multiply $- 1$ by $3.0625$ .

$f' (1.75) = - \frac{4.5}{2 {(20 - 1.75 - 3.0625)}^{\frac{1}{2}}}$

Step 8.2.2.2

Subtract $1.75$ from $20$ .

$f' (1.75) = - \frac{4.5}{2 {(18.25 - 3.0625)}^{\frac{1}{2}}}$

Step 8.2.2.3

Subtract $3.0625$ from $18.25$ .

$f' (1.75) = - \frac{4.5}{2 \cdot {15.1875}^{\frac{1}{2}}}$

Step 8.2.2.4

Rewrite $15.1875$ as ${3.89711431}^{2}$ .

$f' (1.75) = - \frac{4.5}{2 \cdot {({3.89711431}^{2})}^{\frac{1}{2}}}$

Step 8.2.2.5

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

$f' (1.75) = - \frac{4.5}{2 \cdot {3.89711431}^{2 (\frac{1}{2})}}$

Step 8.2.2.6

Cancel the common factor of $2$ .

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Step 8.2.2.6.1

Cancel the common factor.

$f' (1.75) = - \frac{4.5}{2 \cdot {3.89711431}^{2 (\frac{1}{2})}}$

Step 8.2.2.6.2

Rewrite the expression.

$f' (1.75) = - \frac{4.5}{2 \cdot 3.89711431}$

Step 8.2.2.7

Evaluate the exponent.

$f' (1.75) = - \frac{4.5}{2 \cdot 3.89711431}$

Step 8.2.3

Simplify the expression.

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Step 8.2.3.1

Multiply $2$ by $3.89711431$ .

$f' (1.75) = - \frac{4.5}{7.79422863}$

Step 8.2.3.2

Divide $4.5$ by $7.79422863$ .

$f' (1.75) = - 1 \cdot 0.57735026$

Step 8.2.3.3

Multiply $- 1$ by $0.57735026$ .

$f' (1.75) = - 0.57735026$

Step 8.2.4

The final answer is $- 0.57735026$ .

$- 0.57735026$

Step 8.3

At $x = 1.75$ the derivative is $- 0.57735026$ . Since this is negative, the function is decreasing on $(- 0.5, 4)$ .

Decreasing on $(- \frac{1}{2}, 4)$ since $f' (x) < 0$

Step 9

Substitute a value from the interval $(4, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $5$ in the expression.

$f' (5) = - \frac{1 + 2 (5)}{2 {(20 - (5) - {(5)}^{2})}^{\frac{1}{2}}}$

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify the numerator.

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Step 9.2.1.1

Multiply $2$ by $5$ .

$f' (5) = - \frac{1 + 10}{2 {(20 - (5) - 5^{2})}^{\frac{1}{2}}}$

Step 9.2.1.2

Add $1$ and $10$ .

$f' (5) = - \frac{11}{2 {(20 - (5) - 5^{2})}^{\frac{1}{2}}}$

Step 9.2.2

Simplify the denominator.

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Step 9.2.2.1

Simplify each term.

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Step 9.2.2.1.1

Multiply $- 1$ by $5$ .

$f' (5) = - \frac{11}{2 {(20 - 5 - 5^{2})}^{\frac{1}{2}}}$

Step 9.2.2.1.2

Raise $5$ to the power of $2$ .

$f' (5) = - \frac{11}{2 {(20 - 5 - 1 \cdot 25)}^{\frac{1}{2}}}$

Step 9.2.2.1.3

Multiply $- 1$ by $25$ .

$f' (5) = - \frac{11}{2 {(20 - 5 - 25)}^{\frac{1}{2}}}$

Step 9.2.2.2

Subtract $5$ from $20$ .

$f' (5) = - \frac{11}{2 {(15 - 25)}^{\frac{1}{2}}}$

Step 9.2.2.3

Subtract $25$ from $15$ .

$f' (5) = - \frac{11}{2 {(- 10)}^{\frac{1}{2}}}$

Step 9.2.3

The final answer is $- \frac{11}{2 {(- 10)}^{\frac{1}{2}}}$ .

$- \frac{11}{2 {(- 10)}^{\frac{1}{2}}}$

Step 9.3

At $x = 5$ the derivative is $- \frac{11}{2 {(- 10)}^{\frac{1}{2}}}$ . Since this is negative, the function is decreasing on $(4, \infty)$ .

Decreasing on $(4, \infty)$ since $f' (x) < 0$

Step 10

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- 5, - \frac{1}{2})$

Decreasing on: $(- \infty, - 5), (- \frac{1}{2}, 4), (4, \infty)$

Step 11