Calculus Examples

$y = \sqrt{3 + 2 x - x^{2}}$

Step 1

Write $y = \sqrt{3 + 2 x - x^{2}}$ as a function.

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

Step 2

Find the first derivative of the function.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 + 2 x - x^{2}}$ as ${(3 + 2 x - x^{2})}^{\frac{1}{2}}$ .

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

Step 2.2

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 $g (x) = 3 + 2 x - x^{2}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $3 + 2 x - x^{2}$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $3 + 2 x - x^{2}$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.6.2

Subtract $2$ from $1$ .

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

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

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

Step 2.7.2

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

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

Step 2.7.3

Move ${(3 + 2 x - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.8

By the Sum Rule, the derivative of $3 + 2 x - x^{2}$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- x^{2}]$ .

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

Step 2.9

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

Multiply $2$ by $1$ .

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

Step 2.14

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

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

Step 2.15

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

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

Step 2.16

Multiply $2$ by $- 1$ .

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

Step 2.17

Simplify.

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Step 2.17.1

Reorder the factors of $\frac{1}{2 {(3 + 2 x - x^{2})}^{\frac{1}{2}}} (2 - 2 x)$ .

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

Step 2.17.2

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

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

Step 2.17.3

Factor $2$ out of $2$ .

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

Step 2.17.4

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

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

Step 2.17.5

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

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

Step 2.17.6

Cancel the common factors.

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Step 2.17.6.1

Factor $2$ out of $2 {(3 + 2 x - x^{2})}^{\frac{1}{2}}$ .

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

Step 2.17.6.2

Cancel the common factor.

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

Step 2.17.6.3

Rewrite the expression.

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

Step 3

Find the second derivative of the function.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 1 - x$ and $g (x) = {(3 + 2 x - x^{2})}^{\frac{1}{2}}$ .

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

Step 3.2

Multiply the exponents in ${({(3 + 2 x - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor of $2$ .

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Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

Rewrite the expression.

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

Step 3.3

Simplify.

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

Step 3.4

Differentiate.

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Step 3.4.1

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

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

Step 3.4.2

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

$f'' (x) = \frac{{(3 + 2 x - x^{2})}^{\frac{1}{2}} (0 + \frac{d}{d x} (- x)) - (1 - x) \frac{d}{d x} {(3 + 2 x - x^{2})}^{\frac{1}{2}}}{3 + 2 x - x^{2}}$

Step 3.4.3

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

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

Step 3.4.4

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

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

Step 3.4.5

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

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

Step 3.4.6

Simplify the expression.

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Step 3.4.6.1

Multiply $- 1$ by $1$ .

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

Step 3.4.6.2

Move $- 1$ to the left of ${(3 + 2 x - x^{2})}^{\frac{1}{2}}$ .

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

Step 3.4.6.3

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

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

Step 3.5

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 $g (x) = 3 + 2 x - x^{2}$ .

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Step 3.5.1

To apply the Chain Rule, set $u_{1}$ as $3 + 2 x - x^{2}$ .

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

Step 3.5.2

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

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

Step 3.5.3

Replace all occurrences of $u_{1}$ with $3 + 2 x - x^{2}$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Combine the numerators over the common denominator.

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

Step 3.9

Simplify the numerator.

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Step 3.9.1

Multiply $- 1$ by $2$ .

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

Step 3.9.2

Subtract $2$ from $1$ .

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

Step 3.10

Combine fractions.

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Step 3.10.1

Move the negative in front of the fraction.

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

Step 3.10.2

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

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

Step 3.10.3

Move ${(3 + 2 x - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.11

By the Sum Rule, the derivative of $3 + 2 x - x^{2}$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- x^{2}]$ .

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

Step 3.12

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f'' (x) = \frac{- {(3 + 2 x - x^{2})}^{\frac{1}{2}} - (1 - x) (\frac{1}{2 {(3 + 2 x - x^{2})}^{\frac{1}{2}}} \cdot (0 + \frac{d}{d x} (2 x) + \frac{d}{d x} (- x^{2})))}{3 + 2 x - x^{2}}$

Step 3.13

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

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

Step 3.14

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

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

Step 3.15

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

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

Step 3.16

Multiply $2$ by $1$ .

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

Step 3.17

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

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

Step 3.18

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

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

Step 3.19

Multiply $2$ by $- 1$ .

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

Step 3.20

Simplify.

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Step 3.20.1

Apply the distributive property.

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

Step 3.20.2

Simplify the numerator.

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Step 3.20.2.1

Let $u_{2} = {(3 + 2 x - x^{2})}^{\frac{1}{2}}$ . Substitute $u_{2}$ for all occurrences of ${(3 + 2 x - x^{2})}^{\frac{1}{2}}$ .

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

Step 3.20.2.2

Replace all occurrences of $u_{2}$ with ${(3 + 2 x - x^{2})}^{\frac{1}{2}}$ .

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

Step 3.20.2.3

Simplify the numerator.

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Step 3.20.2.3.1

Multiply the exponents in ${({(3 + 2 x - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.20.2.3.1.1

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

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

Step 3.20.2.3.1.2

Cancel the common factor of $2$ .

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Step 3.20.2.3.1.2.1

Cancel the common factor.

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

Step 3.20.2.3.1.2.2

Rewrite the expression.

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

Step 3.20.2.3.2

Simplify.

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

Step 3.20.2.3.3

Add $3$ and $1$ .

$f'' (x) = \frac{- \frac{2 x - x^{2} + x^{2} - 2 x + 4}{{(3 + 2 x - x^{2})}^{\frac{1}{2}}}}{3 + 2 x - x^{2}}$

Step 3.20.2.3.4

Subtract $2 x$ from $2 x$ .

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

Step 3.20.2.3.5

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

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

Step 3.20.2.3.6

Add $0$ and $0$ .

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

Step 3.20.2.3.7

Add $0$ and $4$ .

$f'' (x) = \frac{- \frac{4}{{(3 + 2 x - x^{2})}^{\frac{1}{2}}}}{3 + 2 x - x^{2}}$

Step 3.20.3

Combine terms.

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Step 3.20.3.1

Rewrite $\frac{- \frac{4}{{(3 + 2 x - x^{2})}^{\frac{1}{2}}}}{3 + 2 x - x^{2}}$ as a product.

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

Step 3.20.3.2

Multiply $\frac{1}{3 + 2 x - x^{2}}$ by $\frac{4}{{(3 + 2 x - x^{2})}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{4}{(3 + 2 x - x^{2}) {(3 + 2 x - x^{2})}^{\frac{1}{2}}}$

Step 3.20.3.3

Multiply $3 + 2 x - x^{2}$ by ${(3 + 2 x - x^{2})}^{\frac{1}{2}}$ by adding the exponents.

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Step 3.20.3.3.1

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

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Step 3.20.3.3.1.1

Raise $3 + 2 x - x^{2}$ to the power of $1$ .

$f'' (x) = - \frac{4}{(3 + 2 x - x^{2}) {(3 + 2 x - x^{2})}^{\frac{1}{2}}}$

Step 3.20.3.3.1.2

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

$f'' (x) = - \frac{4}{{(3 + 2 x - x^{2})}^{1 + \frac{1}{2}}}$

Step 3.20.3.3.2

Write $1$ as a fraction with a common denominator.

$f'' (x) = - \frac{4}{{(3 + 2 x - x^{2})}^{\frac{2}{2} + \frac{1}{2}}}$

Step 3.20.3.3.3

Combine the numerators over the common denominator.

$f'' (x) = - \frac{4}{{(3 + 2 x - x^{2})}^{\frac{2 + 1}{2}}}$

Step 3.20.3.3.4

Add $2$ and $1$ .

$f'' (x) = - \frac{4}{{(3 + 2 x - x^{2})}^{\frac{3}{2}}}$

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

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

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 + 2 x - x^{2}}$ as ${(3 + 2 x - x^{2})}^{\frac{1}{2}}$ .

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

Step 5.1.2

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 $g (x) = 3 + 2 x - x^{2}$ .

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Step 5.1.2.1

To apply the Chain Rule, set $u$ as $3 + 2 x - x^{2}$ .

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

Step 5.1.2.2

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

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

Step 5.1.2.3

Replace all occurrences of $u$ with $3 + 2 x - x^{2}$ .

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

Step 5.1.3

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

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

Step 5.1.4

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

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

Step 5.1.5

Combine the numerators over the common denominator.

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

Step 5.1.6

Simplify the numerator.

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Step 5.1.6.1

Multiply $- 1$ by $2$ .

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

Step 5.1.6.2

Subtract $2$ from $1$ .

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

Step 5.1.7

Combine fractions.

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Step 5.1.7.1

Move the negative in front of the fraction.

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

Step 5.1.7.2

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

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

Step 5.1.7.3

Move ${(3 + 2 x - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5.1.8

By the Sum Rule, the derivative of $3 + 2 x - x^{2}$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- x^{2}]$ .

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

Step 5.1.9

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f' (x) = \frac{1}{2 {(3 + 2 x - x^{2})}^{\frac{1}{2}}} \cdot (0 + \frac{d}{d x} (2 x) + \frac{d}{d x} (- x^{2}))$

Step 5.1.10

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

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

Step 5.1.11

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

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

Step 5.1.12

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

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

Step 5.1.13

Multiply $2$ by $1$ .

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

Step 5.1.14

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

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

Step 5.1.15

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

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

Step 5.1.16

Multiply $2$ by $- 1$ .

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

Step 5.1.17

Simplify.

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Step 5.1.17.1

Reorder the factors of $\frac{1}{2 {(3 + 2 x - x^{2})}^{\frac{1}{2}}} (2 - 2 x)$ .

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

Step 5.1.17.2

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

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

Step 5.1.17.3

Factor $2$ out of $2$ .

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

Step 5.1.17.4

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

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

Step 5.1.17.5

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

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

Step 5.1.17.6

Cancel the common factors.

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Step 5.1.17.6.1

Factor $2$ out of $2 {(3 + 2 x - x^{2})}^{\frac{1}{2}}$ .

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

Step 5.1.17.6.2

Cancel the common factor.

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

Step 5.1.17.6.3

Rewrite the expression.

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

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{1 - x}{{(3 + 2 x - x^{2})}^{\frac{1}{2}}}$ .

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

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{1 - x}{{(3 + 2 x - x^{2})}^{\frac{1}{2}}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

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

Step 6.2

Set the numerator equal to zero.

$1 - x = 0$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 6.3.2

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

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Step 6.3.2.1

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

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Dividing two negative values results in a positive value.

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

Step 6.3.2.2.2

Divide $x$ by $1$ .

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

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Convert expressions with fractional exponents to radicals.

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Step 7.1.1

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

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

Step 7.1.2

Anything raised to $1$ is the base itself.

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

Step 7.2

Set the denominator in $\frac{1 - x}{\sqrt{3 + 2 x - x^{2}}}$ equal to $0$ to find where the expression is undefined.

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

Step 7.3

Solve for $x$ .

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Step 7.3.1

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

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

Step 7.3.2

Simplify each side of the equation.

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Step 7.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 + 2 x - x^{2}}$ as ${(3 + 2 x - x^{2})}^{\frac{1}{2}}$ .

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

Step 7.3.2.2

Simplify the left side.

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Step 7.3.2.2.1

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

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Step 7.3.2.2.1.1

Multiply the exponents in ${({(3 + 2 x - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 7.3.2.2.1.1.1

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

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

Step 7.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 7.3.2.2.1.1.2.1

Cancel the common factor.

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

Step 7.3.2.2.1.1.2.2

Rewrite the expression.

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

Step 7.3.2.2.1.2

Simplify.

$3 + 2 x - x^{2} = 0^{2}$

Step 7.3.2.3

Simplify the right side.

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Step 7.3.2.3.1

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

$3 + 2 x - x^{2} = 0$

Step 7.3.3

Solve for $x$ .

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Step 7.3.3.1

Factor the left side of the equation.

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Step 7.3.3.1.1

Factor $- 1$ out of $3 + 2 x - x^{2}$ .

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Step 7.3.3.1.1.1

Reorder the expression.

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Step 7.3.3.1.1.1.1

Move $3$ .

$2 x - x^{2} + 3 = 0$

Step 7.3.3.1.1.1.2

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

$- x^{2} + 2 x + 3 = 0$

Step 7.3.3.1.1.2

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

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

Step 7.3.3.1.1.3

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

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

Step 7.3.3.1.1.4

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 7.3.3.1.1.5

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

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

Step 7.3.3.1.1.6

Factor $- 1$ out of $- (x^{2} - 2 x) - 1 (- 3)$ .

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

Step 7.3.3.1.2

Factor.

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Step 7.3.3.1.2.1

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 7.3.3.1.2.1.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 7.3.3.1.2.1.2

Write the factored form using these integers.

$- ((x - 3) (x + 1)) = 0$

Step 7.3.3.1.2.2

Remove unnecessary parentheses.

$- (x - 3) (x + 1) = 0$

Step 7.3.3.2

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

$x - 3 = 0$

$x + 1 = 0$

Step 7.3.3.3

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 7.3.3.3.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 7.3.3.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 7.3.3.4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 7.3.3.4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 7.3.3.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 7.3.3.5

The final solution is all the values that make $- (x - 3) (x + 1) = 0$ true.

$x = 3, - 1$

Step 7.4

Set the radicand in $\sqrt{3 + 2 x - x^{2}}$ less than $0$ to find where the expression is undefined.

$3 + 2 x - x^{2} < 0$

Step 7.5

Solve for $x$ .

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Step 7.5.1

Convert the inequality to an equation.

$3 + 2 x - x^{2} = 0$

Step 7.5.2

Factor the left side of the equation.

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Step 7.5.2.1

Factor $- 1$ out of $3 + 2 x - x^{2}$ .

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Step 7.5.2.1.1

Reorder the expression.

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Step 7.5.2.1.1.1

Move $3$ .

$2 x - x^{2} + 3 = 0$

Step 7.5.2.1.1.2

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

$- x^{2} + 2 x + 3 = 0$

Step 7.5.2.1.2

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

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

Step 7.5.2.1.3

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

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

Step 7.5.2.1.4

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 7.5.2.1.5

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

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

Step 7.5.2.1.6

Factor $- 1$ out of $- (x^{2} - 2 x) - 1 (- 3)$ .

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

Step 7.5.2.2

Factor.

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Step 7.5.2.2.1

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 7.5.2.2.1.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 7.5.2.2.1.2

Write the factored form using these integers.

$- ((x - 3) (x + 1)) = 0$

Step 7.5.2.2.2

Remove unnecessary parentheses.

$- (x - 3) (x + 1) = 0$

Step 7.5.3

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

$x - 3 = 0$

$x + 1 = 0$

Step 7.5.4

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 7.5.4.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 7.5.4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 7.5.5

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 7.5.5.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 7.5.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 7.5.6

The final solution is all the values that make $- (x - 3) (x + 1) = 0$ true.

$x = 3, - 1$

Step 7.5.7

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 3$

$x > 3$

Step 7.5.8

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 7.5.8.1

Test a value on the interval $x < - 1$ to see if it makes the inequality true.

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Step 7.5.8.1.1

Choose a value on the interval $x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

Step 7.5.8.1.2

Replace $x$ with $- 4$ in the original inequality.

$3 + 2 (- 4) - {(- 4)}^{2} < 0$

Step 7.5.8.1.3

The left side $- 21$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 7.5.8.2

Test a value on the interval $- 1 < x < 3$ to see if it makes the inequality true.

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Step 7.5.8.2.1

Choose a value on the interval $- 1 < x < 3$ and see if this value makes the original inequality true.

$x = 0$

Step 7.5.8.2.2

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

$3 + 2 (0) - {(0)}^{2} < 0$

Step 7.5.8.2.3

The left side $3$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 7.5.8.3

Test a value on the interval $x > 3$ to see if it makes the inequality true.

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Step 7.5.8.3.1

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Step 7.5.8.3.2

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

$3 + 2 (6) - {(6)}^{2} < 0$

Step 7.5.8.3.3

The left side $- 21$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 7.5.8.4

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

$x < - 1$ True

$- 1 < x < 3$ False

$x > 3$ True

$x < - 1$ True

$- 1 < x < 3$ False

$x > 3$ True

Step 7.5.9

The solution consists of all of the true intervals.

$x < - 1$ or $x > 3$

Step 7.6

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 $0$ .

$x \leq - 1, x \geq 3$

$(- \infty, - 1] \cup [3, \infty)$

$x \leq - 1, x \geq 3$

$(- \infty, - 1] \cup [3, \infty)$

Step 8

Critical points to evaluate.

$x = 1, - 1, 3$

Step 9

Evaluate the second derivative at $x = 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{4}{{(3 + 2 (1) - {(1)}^{2})}^{\frac{3}{2}}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the denominator.

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Step 10.1.1

Simplify each term.

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Step 10.1.1.1

Multiply $2$ by $1$ .

$- \frac{4}{{(3 + 2 - 1^{2})}^{\frac{3}{2}}}$

Step 10.1.1.2

One to any power is one.

$- \frac{4}{{(3 + 2 - 1 \cdot 1)}^{\frac{3}{2}}}$

Step 10.1.1.3

Multiply $- 1$ by $1$ .

$- \frac{4}{{(3 + 2 - 1)}^{\frac{3}{2}}}$

Step 10.1.2

Add $3$ and $2$ .

$- \frac{4}{{(5 - 1)}^{\frac{3}{2}}}$

Step 10.1.3

Subtract $1$ from $5$ .

$- \frac{4}{4^{\frac{3}{2}}}$

Step 10.1.4

Rewrite $4$ as $2^{2}$ .

$- \frac{4}{{(2^{2})}^{\frac{3}{2}}}$

Step 10.1.5

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

$- \frac{4}{2^{2 (\frac{3}{2})}}$

Step 10.1.6

Cancel the common factor of $2$ .

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Step 10.1.6.1

Cancel the common factor.

$- \frac{4}{2^{2 (\frac{3}{2})}}$

Step 10.1.6.2

Rewrite the expression.

$- \frac{4}{2^{3}}$

Step 10.1.7

Raise $2$ to the power of $3$ .

$- \frac{4}{8}$

Step 10.2

Cancel the common factor of $4$ and $8$ .

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Step 10.2.1

Factor $4$ out of $4$ .

$- \frac{4 (1)}{8}$

Step 10.2.2

Cancel the common factors.

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Step 10.2.2.1

Factor $4$ out of $8$ .

$- \frac{4 \cdot 1}{4 \cdot 2}$

Step 10.2.2.2

Cancel the common factor.

$- \frac{4 \cdot 1}{4 \cdot 2}$

Step 10.2.2.3

Rewrite the expression.

$- \frac{1}{2}$

Step 11

$x = 1$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 1$ is a local maximum

Step 12

Find the y-value when $x = 1$ .

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Step 12.1

Replace the variable $x$ with $1$ in the expression.

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

Step 12.2

Simplify the result.

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Step 12.2.1

Multiply $2$ by $1$ .

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

Step 12.2.2

One to any power is one.

$f (1) = \sqrt{3 + 2 - 1 \cdot 1}$

Step 12.2.3

Multiply $- 1$ by $1$ .

$f (1) = \sqrt{3 + 2 - 1}$

Step 12.2.4

Add $3$ and $2$ .

$f (1) = \sqrt{5 - 1}$

Step 12.2.5

Subtract $1$ from $5$ .

$f (1) = \sqrt{4}$

Step 12.2.6

Rewrite $4$ as $2^{2}$ .

$f (1) = \sqrt{2^{2}}$

Step 12.2.7

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

$f (1) = 2$

Step 12.2.8

The final answer is $2$ .

$y = 2$

Step 13

Evaluate the second derivative at $x = - 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{4}{{(3 + 2 (- 1) - {(- 1)}^{2})}^{\frac{3}{2}}}$

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify each term.

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Step 14.1.1

Multiply $2$ by $- 1$ .

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

Step 14.1.2

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

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Step 14.1.2.1

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

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Step 14.1.2.1.1

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

$- \frac{4}{{(3 - 2 + {(- 1)}^{1} {(- 1)}^{2})}^{\frac{3}{2}}}$

Step 14.1.2.1.2

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

$- \frac{4}{{(3 - 2 + {(- 1)}^{1 + 2})}^{\frac{3}{2}}}$

Step 14.1.2.2

Add $1$ and $2$ .

$- \frac{4}{{(3 - 2 + {(- 1)}^{3})}^{\frac{3}{2}}}$

Step 14.1.3

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

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

Step 14.2

Reduce the expression by cancelling the common factors.

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Step 14.2.1

Subtract $2$ from $3$ .

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

Step 14.2.2

Simplify the expression.

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Step 14.2.2.1

Subtract $1$ from $1$ .

$- \frac{4}{0^{\frac{3}{2}}}$

Step 14.2.2.2

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

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

Step 14.2.2.3

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

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

Step 14.2.3

Cancel the common factor of $2$ .

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Step 14.2.3.1

Cancel the common factor.

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

Step 14.2.3.2

Rewrite the expression.

$- \frac{4}{0^{3}}$

Step 14.2.4

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

$- \frac{4}{0}$

Step 14.2.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$- \frac{4}{0}$

Step 14.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 15

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 16