Calculus Examples

$f (x) = 9 \sin (x) \cos (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

Since $9$ is constant with respect to $x$ , the derivative of $9 \sin (x) \cos (x)$ with respect to $x$ is $9 \frac{d}{d x} [\sin (x) \cos (x)]$ .

$9 \frac{d}{d x} [\sin (x) \cos (x)]$

Step 1.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sin (x)$ and $g (x) = \cos (x)$ .

$9 (\sin (x) \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 1.3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$9 (\sin (x) (- \sin (x)) + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 1.4

Raise $\sin (x)$ to the power of $1$ .

$9 (- (\sin^{1} (x) \sin (x)) + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 1.5

Raise $\sin (x)$ to the power of $1$ .

$9 (- (\sin^{1} (x) \sin^{1} (x)) + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 1.6

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

$9 (- {\sin (x)}^{1 + 1} + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 1.7

Add $1$ and $1$ .

$9 (- \sin^{2} (x) + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 1.8

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$9 (- \sin^{2} (x) + \cos (x) \cos (x))$

Step 1.9

Raise $\cos (x)$ to the power of $1$ .

$9 (- \sin^{2} (x) + \cos^{1} (x) \cos (x))$

Step 1.10

Raise $\cos (x)$ to the power of $1$ .

$9 (- \sin^{2} (x) + \cos^{1} (x) \cos^{1} (x))$

Step 1.11

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

$9 (- \sin^{2} (x) + {\cos (x)}^{1 + 1})$

Step 1.12

Add $1$ and $1$ .

$9 (- \sin^{2} (x) + \cos^{2} (x))$

Step 1.13

Simplify.

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Step 1.13.1

Apply the distributive property.

$9 (- \sin^{2} (x)) + 9 \cos^{2} (x)$

Step 1.13.2

Multiply $- 1$ by $9$ .

$- 9 \sin^{2} (x) + 9 \cos^{2} (x)$

Step 1.13.3

Rewrite $9 \cos^{2} (x)$ as ${(3 \cos (x))}^{2}$ .

$- 9 \sin^{2} (x) + {(3 \cos (x))}^{2}$

Step 1.13.4

Rewrite $9 \sin^{2} (x)$ as ${(3 \sin (x))}^{2}$ .

$- {(3 \sin (x))}^{2} + {(3 \cos (x))}^{2}$

Step 1.13.5

Reorder $- {(3 \sin (x))}^{2}$ and ${(3 \cos (x))}^{2}$ .

${(3 \cos (x))}^{2} - {(3 \sin (x))}^{2}$

Step 1.13.6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 \cos (x)$ and $b = 3 \sin (x)$ .

$(3 \cos (x) + 3 \sin (x)) (3 \cos (x) - (3 \sin (x)))$

Step 1.13.7

Multiply $3$ by $- 1$ .

$(3 \cos (x) + 3 \sin (x)) (3 \cos (x) - 3 \sin (x))$

Step 1.13.8

Expand $(3 \cos (x) + 3 \sin (x)) (3 \cos (x) - 3 \sin (x))$ using the FOIL Method.

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Step 1.13.8.1

Apply the distributive property.

$3 \cos (x) (3 \cos (x) - 3 \sin (x)) + 3 \sin (x) (3 \cos (x) - 3 \sin (x))$

Step 1.13.8.2

Apply the distributive property.

$3 \cos (x) (3 \cos (x)) + 3 \cos (x) (- 3 \sin (x)) + 3 \sin (x) (3 \cos (x) - 3 \sin (x))$

Step 1.13.8.3

Apply the distributive property.

$3 \cos (x) (3 \cos (x)) + 3 \cos (x) (- 3 \sin (x)) + 3 \sin (x) (3 \cos (x)) + 3 \sin (x) (- 3 \sin (x))$

Step 1.13.9

Combine the opposite terms in $3 \cos (x) (3 \cos (x)) + 3 \cos (x) (- 3 \sin (x)) + 3 \sin (x) (3 \cos (x)) + 3 \sin (x) (- 3 \sin (x))$ .

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Step 1.13.9.1

Reorder the factors in the terms $3 \cos (x) (- 3 \sin (x))$ and $3 \sin (x) (3 \cos (x))$ .

$3 \cos (x) (3 \cos (x)) - 3 \cdot 3 \cos (x) \sin (x) + 3 \cdot 3 \cos (x) \sin (x) + 3 \sin (x) (- 3 \sin (x))$

Step 1.13.9.2

Add $- 3 \cdot 3 \cos (x) \sin (x)$ and $3 \cdot 3 \cos (x) \sin (x)$ .

$3 \cos (x) (3 \cos (x)) + 0 + 3 \sin (x) (- 3 \sin (x))$

Step 1.13.9.3

Add $3 \cos (x) (3 \cos (x))$ and $0$ .

$3 \cos (x) (3 \cos (x)) + 3 \sin (x) (- 3 \sin (x))$

Step 1.13.10

Simplify each term.

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Step 1.13.10.1

Multiply $3 \cos (x) (3 \cos (x))$ .

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Step 1.13.10.1.1

Multiply $3$ by $3$ .

$9 \cos (x) \cos (x) + 3 \sin (x) (- 3 \sin (x))$

Step 1.13.10.1.2

Raise $\cos (x)$ to the power of $1$ .

$9 (\cos^{1} (x) \cos (x)) + 3 \sin (x) (- 3 \sin (x))$

Step 1.13.10.1.3

Raise $\cos (x)$ to the power of $1$ .

$9 (\cos^{1} (x) \cos^{1} (x)) + 3 \sin (x) (- 3 \sin (x))$

Step 1.13.10.1.4

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

$9 {\cos (x)}^{1 + 1} + 3 \sin (x) (- 3 \sin (x))$

Step 1.13.10.1.5

Add $1$ and $1$ .

$9 \cos^{2} (x) + 3 \sin (x) (- 3 \sin (x))$

Step 1.13.10.2

Multiply $3 \sin (x) (- 3 \sin (x))$ .

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Step 1.13.10.2.1

Multiply $- 3$ by $3$ .

$9 \cos^{2} (x) - 9 \sin (x) \sin (x)$

Step 1.13.10.2.2

Raise $\sin (x)$ to the power of $1$ .

$9 \cos^{2} (x) - 9 (\sin^{1} (x) \sin (x))$

Step 1.13.10.2.3

Raise $\sin (x)$ to the power of $1$ .

$9 \cos^{2} (x) - 9 (\sin^{1} (x) \sin^{1} (x))$

Step 1.13.10.2.4

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

$9 \cos^{2} (x) - 9 {\sin (x)}^{1 + 1}$

Step 1.13.10.2.5

Add $1$ and $1$ .

$9 \cos^{2} (x) - 9 \sin^{2} (x)$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $9 \cos^{2} (x) - 9 \sin^{2} (x)$ with respect to $x$ is $\frac{d}{d x} [9 \cos^{2} (x)] + \frac{d}{d x} [- 9 \sin^{2} (x)]$ .

$f'' (x) = \frac{d}{d x} (9 \cos^{2} (x)) + \frac{d}{d x} (- 9 \sin^{2} (x))$

Step 2.2

Evaluate $\frac{d}{d x} [9 \cos^{2} (x)]$ .

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Step 2.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 \cos^{2} (x)$ with respect to $x$ is $9 \frac{d}{d x} [\cos^{2} (x)]$ .

$f'' (x) = 9 \frac{d}{d x} (\cos^{2} (x)) + \frac{d}{d x} (- 9 \sin^{2} (x))$

Step 2.2.2

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

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Step 2.2.2.1

To apply the Chain Rule, set $u_{1}$ as $\cos (x)$ .

$f'' (x) = 9 (\frac{d}{d u (1)} ({u_{1}}^{2}) \frac{d}{d x} (\cos (x))) + \frac{d}{d x} (- 9 \sin^{2} (x))$

Step 2.2.2.2

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

$f'' (x) = 9 (2 u_{1} \frac{d}{d x} (\cos (x))) + \frac{d}{d x} (- 9 \sin^{2} (x))$

Step 2.2.2.3

Replace all occurrences of $u_{1}$ with $\cos (x)$ .

$f'' (x) = 9 (2 \cos (x) \frac{d}{d x} (\cos (x))) + \frac{d}{d x} (- 9 \sin^{2} (x))$

Step 2.2.3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$f'' (x) = 9 (2 \cos (x) (- \sin (x))) + \frac{d}{d x} (- 9 \sin^{2} (x))$

Step 2.2.4

Multiply $- 1$ by $2$ .

$f'' (x) = 9 (- 2 \cos (x) \sin (x)) + \frac{d}{d x} (- 9 \sin^{2} (x))$

Step 2.2.5

Multiply $- 2$ by $9$ .

$f'' (x) = - 18 \cos (x) \sin (x) + \frac{d}{d x} (- 9 \sin^{2} (x))$

Step 2.3

Evaluate $\frac{d}{d x} [- 9 \sin^{2} (x)]$ .

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Step 2.3.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 \sin^{2} (x)$ with respect to $x$ is $- 9 \frac{d}{d x} [\sin^{2} (x)]$ .

$f'' (x) = - 18 \cos (x) \sin (x) - 9 \frac{d}{d x} \sin^{2} (x)$

Step 2.3.2

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

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Step 2.3.2.1

To apply the Chain Rule, set $u_{2}$ as $\sin (x)$ .

$f'' (x) = - 18 \cos (x) \sin (x) - 9 (\frac{d}{d u (2)} ({u_{2}}^{2}) \frac{d}{d x} (\sin (x)))$

Step 2.3.2.2

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

$f'' (x) = - 18 \cos (x) \sin (x) - 9 (2 u_{2} \frac{d}{d x} (\sin (x)))$

Step 2.3.2.3

Replace all occurrences of $u_{2}$ with $\sin (x)$ .

$f'' (x) = - 18 \cos (x) \sin (x) - 9 (2 \sin (x) \frac{d}{d x} (\sin (x)))$

Step 2.3.3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$f'' (x) = - 18 \cos (x) \sin (x) - 9 (2 \sin (x) \cos (x))$

Step 2.3.4

Multiply $2$ by $- 9$ .

$f'' (x) = - 18 \cos (x) \sin (x) - 18 \sin (x) \cos (x)$

Step 2.4

Combine terms.

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Step 2.4.1

Reorder the factors of $- 18 \sin (x) \cos (x)$ .

$f'' (x) = - 18 \cos (x) \sin (x) - 18 \cos (x) \sin (x)$

Step 2.4.2

Subtract $18 \cos (x) \sin (x)$ from $- 18 \cos (x) \sin (x)$ .

$f'' (x) = - 36 \cos (x) \sin (x)$

Step 3

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

$9 \cos^{2} (x) - 9 \sin^{2} (x) = 0$

Step 4

Factor $9 \cos^{2} (x) - 9 \sin^{2} (x)$ .

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Step 4.1

Factor $9$ out of $9 \cos^{2} (x) - 9 \sin^{2} (x)$ .

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Step 4.1.1

Factor $9$ out of $9 \cos^{2} (x)$ .

$9 \cos^{2} (x) - 9 \sin^{2} (x) = 0$

Step 4.1.2

Factor $9$ out of $- 9 \sin^{2} (x)$ .

$9 \cos^{2} (x) + 9 (- \sin^{2} (x)) = 0$

Step 4.1.3

Factor $9$ out of $9 \cos^{2} (x) + 9 (- \sin^{2} (x))$ .

$9 (\cos^{2} (x) - \sin^{2} (x)) = 0$

Step 4.2

Factor.

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Step 4.2.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (x)$ and $b = \sin (x)$ .

$9 ((\cos (x) + \sin (x)) (\cos (x) - \sin (x))) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$9 (\cos (x) + \sin (x)) (\cos (x) - \sin (x)) = 0$

Step 5

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

$\cos (x) + \sin (x) = 0$

$\cos (x) - \sin (x) = 0$

Step 6

Set $\cos (x) + \sin (x)$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $\cos (x) + \sin (x)$ equal to $0$ .

$\cos (x) + \sin (x) = 0$

Step 6.2

Solve $\cos (x) + \sin (x) = 0$ for $x$ .

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Step 6.2.1

Divide each term in the equation by $\cos (x)$ .

$\frac{\cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 6.2.2

Cancel the common factor of $\cos (x)$ .

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Step 6.2.2.1

Cancel the common factor.

$\frac{\cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 6.2.2.2

Rewrite the expression.

$1 + \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 6.2.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$1 + \tan (x) = \frac{0}{\cos (x)}$

Step 6.2.4

Separate fractions.

$1 + \tan (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 6.2.5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$1 + \tan (x) = \frac{0}{1} \cdot \sec (x)$

Step 6.2.6

Divide $0$ by $1$ .

$1 + \tan (x) = 0 \sec (x)$

Step 6.2.7

Multiply $0$ by $\sec (x)$ .

$1 + \tan (x) = 0$

Step 6.2.8

Subtract $1$ from both sides of the equation.

$\tan (x) = - 1$

Step 6.2.9

Take the inverse tangent of both sides of the equation to extract $x$ from inside the tangent.

$x = \arctan (- 1)$

Step 6.2.10

Simplify the right side.

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Step 6.2.10.1

The exact value of $\arctan (- 1)$ is $- \frac{π}{4}$ .

$x = - \frac{π}{4}$

Step 6.2.11

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the third quadrant.

$x = - \frac{π}{4} - π$

Step 6.2.12

Simplify the expression to find the second solution.

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Step 6.2.12.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

Step 6.2.12.2

The resulting angle of $\frac{3 π}{4}$ is positive and coterminal with $- \frac{π}{4} - π$ .

$x = \frac{3 π}{4}$

Step 6.2.13

The solution to the equation $x = - \frac{π}{4}$ .

$x = - \frac{π}{4}, \frac{3 π}{4}$

Step 7

Set $\cos (x) - \sin (x)$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $\cos (x) - \sin (x)$ equal to $0$ .

$\cos (x) - \sin (x) = 0$

Step 7.2

Solve $\cos (x) - \sin (x) = 0$ for $x$ .

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Step 7.2.1

Divide each term in the equation by $\cos (x)$ .

$\frac{\cos (x)}{\cos (x)} + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 7.2.2

Cancel the common factor of $\cos (x)$ .

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Step 7.2.2.1

Cancel the common factor.

$\frac{\cos (x)}{\cos (x)} + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 7.2.2.2

Rewrite the expression.

$1 + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 7.2.3

Separate fractions.

$1 + \frac{- 1}{1} \cdot \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 7.2.4

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$1 + \frac{- 1}{1} \cdot \tan (x) = \frac{0}{\cos (x)}$

Step 7.2.5

Divide $- 1$ by $1$ .

$1 - \tan (x) = \frac{0}{\cos (x)}$

Step 7.2.6

Separate fractions.

$1 - \tan (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 7.2.7

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$1 - \tan (x) = \frac{0}{1} \cdot \sec (x)$

Step 7.2.8

Divide $0$ by $1$ .

$1 - \tan (x) = 0 \sec (x)$

Step 7.2.9

Multiply $0$ by $\sec (x)$ .

$1 - \tan (x) = 0$

Step 7.2.10

Subtract $1$ from both sides of the equation.

$- \tan (x) = - 1$

Step 7.2.11

Divide each term in $- \tan (x) = - 1$ by $- 1$ and simplify.

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Step 7.2.11.1

Divide each term in $- \tan (x) = - 1$ by $- 1$ .

$\frac{- \tan (x)}{- 1} = \frac{- 1}{- 1}$

Step 7.2.11.2

Simplify the left side.

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Step 7.2.11.2.1

Dividing two negative values results in a positive value.

$\frac{\tan (x)}{1} = \frac{- 1}{- 1}$

Step 7.2.11.2.2

Divide $\tan (x)$ by $1$ .

$\tan (x) = \frac{- 1}{- 1}$

Step 7.2.11.3

Simplify the right side.

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Step 7.2.11.3.1

Divide $- 1$ by $- 1$ .

$\tan (x) = 1$

Step 7.2.12

Take the inverse tangent of both sides of the equation to extract $x$ from inside the tangent.

$x = \arctan (1)$

Step 7.2.13

Simplify the right side.

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Step 7.2.13.1

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 7.2.14

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = π + \frac{π}{4}$

Step 7.2.15

Simplify $π + \frac{π}{4}$ .

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Step 7.2.15.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 7.2.15.2

Combine fractions.

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Step 7.2.15.2.1

Combine $π$ and $\frac{4}{4}$ .

$x = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 7.2.15.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 7.2.15.3

Simplify the numerator.

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Step 7.2.15.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 7.2.15.3.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Step 7.2.16

The solution to the equation $x = \frac{π}{4}$ .

$x = \frac{π}{4}, \frac{5 π}{4}$

Step 8

The final solution is all the values that make $9 (\cos (x) + \sin (x)) (\cos (x) - \sin (x)) = 0$ true.

$x = - \frac{π}{4}, \frac{3 π}{4}, \frac{π}{4}, \frac{5 π}{4}$

Step 9

Evaluate the second derivative at $x = - \frac{π}{4}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 36 \cos (- \frac{π}{4}) \sin (- \frac{π}{4})$

Step 10

Evaluate the second derivative.

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Step 10.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$- 36 \cos (\frac{7 π}{4}) \sin (- \frac{π}{4})$

Step 10.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$- 36 \cos (\frac{π}{4}) \sin (- \frac{π}{4})$

Step 10.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- 36 \frac{\sqrt{2}}{2} \sin (- \frac{π}{4})$

Step 10.4

Cancel the common factor of $2$ .

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Step 10.4.1

Factor $2$ out of $- 36$ .

$2 (- 18) \frac{\sqrt{2}}{2} \sin (- \frac{π}{4})$

Step 10.4.2

Cancel the common factor.

$2 \cdot - 18 \frac{\sqrt{2}}{2} \sin (- \frac{π}{4})$

Step 10.4.3

Rewrite the expression.

$- 18 \sqrt{2} \sin (- \frac{π}{4})$

Step 10.5

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$- 18 \sqrt{2} \sin (\frac{7 π}{4})$

Step 10.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$- 18 \sqrt{2} (- \sin (\frac{π}{4}))$

Step 10.7

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- 18 \sqrt{2} (- \frac{\sqrt{2}}{2})$

Step 10.8

Cancel the common factor of $2$ .

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Step 10.8.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

$- 18 \sqrt{2} \frac{- \sqrt{2}}{2}$

Step 10.8.2

Factor $2$ out of $- 18 \sqrt{2}$ .

$2 (- 9 \sqrt{2}) \frac{- \sqrt{2}}{2}$

Step 10.8.3

Cancel the common factor.

$2 (- 9 \sqrt{2}) \frac{- \sqrt{2}}{2}$

Step 10.8.4

Rewrite the expression.

$- 9 \sqrt{2} (- \sqrt{2})$

Step 10.9

Multiply $- 1$ by $- 9$ .

$9 \sqrt{2} \sqrt{2}$

Step 10.10

Raise $\sqrt{2}$ to the power of $1$ .

$9 ({\sqrt{2}}^{1} \sqrt{2})$

Step 10.11

Raise $\sqrt{2}$ to the power of $1$ .

$9 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

Step 10.12

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

$9 {\sqrt{2}}^{1 + 1}$

Step 10.13

Add $1$ and $1$ .

$9 {\sqrt{2}}^{2}$

Step 10.14

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

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Step 10.14.1

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

$9 {(2^{\frac{1}{2}})}^{2}$

Step 10.14.2

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

$9 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 10.14.3

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

$9 \cdot 2^{\frac{2}{2}}$

Step 10.14.4

Cancel the common factor of $2$ .

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Step 10.14.4.1

Cancel the common factor.

$9 \cdot 2^{\frac{2}{2}}$

Step 10.14.4.2

Rewrite the expression.

$9 \cdot 2^{1}$

Step 10.14.5

Evaluate the exponent.

$9 \cdot 2$

Step 10.15

Multiply $9$ by $2$ .

$18$

Step 11

$x = - \frac{π}{4}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{π}{4}$ is a local minimum

Step 12

Find the y-value when $x = - \frac{π}{4}$ .

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Step 12.1

Replace the variable $x$ with $- \frac{π}{4}$ in the expression.

$f (- \frac{π}{4}) = 9 \sin (- \frac{π}{4}) \cos (- \frac{π}{4})$

Step 12.2

Simplify the result.

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Step 12.2.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (- \frac{π}{4}) = 9 \sin (\frac{7 π}{4}) \cos (- \frac{π}{4})$

Step 12.2.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$f (- \frac{π}{4}) = 9 (- \sin (\frac{π}{4})) \cos (- \frac{π}{4})$

Step 12.2.3

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$f (- \frac{π}{4}) = 9 (- \frac{\sqrt{2}}{2}) \cdot \cos (- \frac{π}{4})$

Step 12.2.4

Multiply $9 (- \frac{\sqrt{2}}{2})$ .

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Step 12.2.4.1

Multiply $- 1$ by $9$ .

$f (- \frac{π}{4}) = - 9 \frac{\sqrt{2}}{2} \cdot \cos (- \frac{π}{4})$

Step 12.2.4.2

Combine $- 9$ and $\frac{\sqrt{2}}{2}$ .

$f (- \frac{π}{4}) = \frac{- 9 \sqrt{2}}{2} \cdot \cos (- \frac{π}{4})$

Step 12.2.5

Move the negative in front of the fraction.

$f (- \frac{π}{4}) = - \frac{9 \sqrt{2}}{2} \cdot \cos (- \frac{π}{4})$

Step 12.2.6

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (- \frac{π}{4}) = - \frac{9 \sqrt{2}}{2} \cdot \cos (\frac{7 π}{4})$

Step 12.2.7

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$f (- \frac{π}{4}) = - \frac{9 \sqrt{2}}{2} \cdot \cos (\frac{π}{4})$

Step 12.2.8

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$f (- \frac{π}{4}) = - \frac{9 \sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}$

Step 12.2.9

Multiply $- \frac{9 \sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 12.2.9.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{9 \sqrt{2}}{2}$ .

$f (- \frac{π}{4}) = - \frac{\sqrt{2} (9 \sqrt{2})}{2 \cdot 2}$

Step 12.2.9.2

Raise $\sqrt{2}$ to the power of $1$ .

$f (- \frac{π}{4}) = - \frac{9 (\sqrt{2} \sqrt{2})}{2 \cdot 2}$

Step 12.2.9.3

Raise $\sqrt{2}$ to the power of $1$ .

$f (- \frac{π}{4}) = - \frac{9 (\sqrt{2} \sqrt{2})}{2 \cdot 2}$

Step 12.2.9.4

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

$f (- \frac{π}{4}) = - \frac{9 {\sqrt{2}}^{1 + 1}}{2 \cdot 2}$

Step 12.2.9.5

Add $1$ and $1$ .

$f (- \frac{π}{4}) = - \frac{9 {\sqrt{2}}^{2}}{2 \cdot 2}$

Step 12.2.9.6

Multiply $2$ by $2$ .

$f (- \frac{π}{4}) = - \frac{9 {\sqrt{2}}^{2}}{4}$

Step 12.2.10

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

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Step 12.2.10.1

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

$f (- \frac{π}{4}) = - \frac{9 {(2^{\frac{1}{2}})}^{2}}{4}$

Step 12.2.10.2

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

$f (- \frac{π}{4}) = - \frac{9 \cdot 2^{\frac{1}{2} \cdot 2}}{4}$

Step 12.2.10.3

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

$f (- \frac{π}{4}) = - \frac{9 \cdot 2^{\frac{2}{2}}}{4}$

Step 12.2.10.4

Cancel the common factor of $2$ .

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Step 12.2.10.4.1

Cancel the common factor.

$f (- \frac{π}{4}) = - \frac{9 \cdot 2^{\frac{2}{2}}}{4}$

Step 12.2.10.4.2

Rewrite the expression.

$f (- \frac{π}{4}) = - \frac{9 \cdot 2}{4}$

Step 12.2.10.5

Evaluate the exponent.

$f (- \frac{π}{4}) = - \frac{9 \cdot 2}{4}$

Step 12.2.11

Multiply $9$ by $2$ .

$f (- \frac{π}{4}) = - \frac{18}{4}$

Step 12.2.12

Cancel the common factor of $18$ and $4$ .

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Step 12.2.12.1

Factor $2$ out of $18$ .

$f (- \frac{π}{4}) = - \frac{2 (9)}{4}$

Step 12.2.12.2

Cancel the common factors.

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Step 12.2.12.2.1

Factor $2$ out of $4$ .

$f (- \frac{π}{4}) = - \frac{2 \cdot 9}{2 \cdot 2}$

Step 12.2.12.2.2

Cancel the common factor.

$f (- \frac{π}{4}) = - \frac{2 \cdot 9}{2 \cdot 2}$

Step 12.2.12.2.3

Rewrite the expression.

$f (- \frac{π}{4}) = - \frac{9}{2}$

Step 12.2.13

The final answer is $- \frac{9}{2}$ .

$y = - \frac{9}{2}$

Step 13

Evaluate the second derivative at $x = \frac{3 π}{4}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 36 \cos (\frac{3 π}{4}) \sin (\frac{3 π}{4})$

Step 14

Evaluate the second derivative.

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Step 14.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$- 36 (- \cos (\frac{π}{4})) \sin (\frac{3 π}{4})$

Step 14.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- 36 (- \frac{\sqrt{2}}{2}) \sin (\frac{3 π}{4})$

Step 14.3

Cancel the common factor of $2$ .

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Step 14.3.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

$- 36 \frac{- \sqrt{2}}{2} \sin (\frac{3 π}{4})$

Step 14.3.2

Factor $2$ out of $- 36$ .

$2 (- 18) \frac{- \sqrt{2}}{2} \sin (\frac{3 π}{4})$

Step 14.3.3

Cancel the common factor.

$2 \cdot - 18 \frac{- \sqrt{2}}{2} \sin (\frac{3 π}{4})$

Step 14.3.4

Rewrite the expression.

$- 18 (- \sqrt{2}) \sin (\frac{3 π}{4})$

Step 14.4

Multiply $- 1$ by $- 18$ .

$18 \sqrt{2} \sin (\frac{3 π}{4})$

Step 14.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$18 \sqrt{2} \sin (\frac{π}{4})$

Step 14.6

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$18 \sqrt{2} \frac{\sqrt{2}}{2}$

Step 14.7

Cancel the common factor of $2$ .

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Step 14.7.1

Factor $2$ out of $18 \sqrt{2}$ .

$2 (9 \sqrt{2}) \frac{\sqrt{2}}{2}$

Step 14.7.2

Cancel the common factor.

$2 (9 \sqrt{2}) \frac{\sqrt{2}}{2}$

Step 14.7.3

Rewrite the expression.

$9 \sqrt{2} \sqrt{2}$

Step 14.8

Raise $\sqrt{2}$ to the power of $1$ .

$9 ({\sqrt{2}}^{1} \sqrt{2})$

Step 14.9

Raise $\sqrt{2}$ to the power of $1$ .

$9 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

Step 14.10

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

$9 {\sqrt{2}}^{1 + 1}$

Step 14.11

Add $1$ and $1$ .

$9 {\sqrt{2}}^{2}$

Step 14.12

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

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Step 14.12.1

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

$9 {(2^{\frac{1}{2}})}^{2}$

Step 14.12.2

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

$9 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 14.12.3

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

$9 \cdot 2^{\frac{2}{2}}$

Step 14.12.4

Cancel the common factor of $2$ .

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Step 14.12.4.1

Cancel the common factor.

$9 \cdot 2^{\frac{2}{2}}$

Step 14.12.4.2

Rewrite the expression.

$9 \cdot 2^{1}$

Step 14.12.5

Evaluate the exponent.

$9 \cdot 2$

Step 14.13

Multiply $9$ by $2$ .

$18$

Step 15

$x = \frac{3 π}{4}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{3 π}{4}$ is a local minimum

Step 16

Find the y-value when $x = \frac{3 π}{4}$ .

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Step 16.1

Replace the variable $x$ with $\frac{3 π}{4}$ in the expression.

$f (\frac{3 π}{4}) = 9 \sin (\frac{3 π}{4}) \cos (\frac{3 π}{4})$

Step 16.2

Simplify the result.

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Step 16.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$f (\frac{3 π}{4}) = 9 \sin (\frac{π}{4}) \cos (\frac{3 π}{4})$

Step 16.2.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$f (\frac{3 π}{4}) = 9 (\frac{\sqrt{2}}{2}) \cdot \cos (\frac{3 π}{4})$

Step 16.2.3

Combine $9$ and $\frac{\sqrt{2}}{2}$ .

$f (\frac{3 π}{4}) = \frac{9 \sqrt{2}}{2} \cdot \cos (\frac{3 π}{4})$

Step 16.2.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$f (\frac{3 π}{4}) = \frac{9 \sqrt{2}}{2} \cdot (- \cos (\frac{π}{4}))$

Step 16.2.5

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$f (\frac{3 π}{4}) = \frac{9 \sqrt{2}}{2} \cdot (- \frac{\sqrt{2}}{2})$

Step 16.2.6

Multiply $\frac{9 \sqrt{2}}{2} (- \frac{\sqrt{2}}{2})$ .

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Step 16.2.6.1

Multiply $\frac{9 \sqrt{2}}{2}$ by $\frac{\sqrt{2}}{2}$ .

$f (\frac{3 π}{4}) = - \frac{9 \sqrt{2} \sqrt{2}}{2 \cdot 2}$

Step 16.2.6.2

Raise $\sqrt{2}$ to the power of $1$ .

$f (\frac{3 π}{4}) = - \frac{9 (\sqrt{2} \sqrt{2})}{2 \cdot 2}$

Step 16.2.6.3

Raise $\sqrt{2}$ to the power of $1$ .

$f (\frac{3 π}{4}) = - \frac{9 (\sqrt{2} \sqrt{2})}{2 \cdot 2}$

Step 16.2.6.4

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

$f (\frac{3 π}{4}) = - \frac{9 {\sqrt{2}}^{1 + 1}}{2 \cdot 2}$

Step 16.2.6.5

Add $1$ and $1$ .

$f (\frac{3 π}{4}) = - \frac{9 {\sqrt{2}}^{2}}{2 \cdot 2}$

Step 16.2.6.6

Multiply $2$ by $2$ .

$f (\frac{3 π}{4}) = - \frac{9 {\sqrt{2}}^{2}}{4}$

Step 16.2.7

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

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Step 16.2.7.1

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

$f (\frac{3 π}{4}) = - \frac{9 {(2^{\frac{1}{2}})}^{2}}{4}$

Step 16.2.7.2

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

$f (\frac{3 π}{4}) = - \frac{9 \cdot 2^{\frac{1}{2} \cdot 2}}{4}$

Step 16.2.7.3

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

$f (\frac{3 π}{4}) = - \frac{9 \cdot 2^{\frac{2}{2}}}{4}$

Step 16.2.7.4

Cancel the common factor of $2$ .

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Step 16.2.7.4.1

Cancel the common factor.

$f (\frac{3 π}{4}) = - \frac{9 \cdot 2^{\frac{2}{2}}}{4}$

Step 16.2.7.4.2

Rewrite the expression.

$f (\frac{3 π}{4}) = - \frac{9 \cdot 2}{4}$

Step 16.2.7.5

Evaluate the exponent.

$f (\frac{3 π}{4}) = - \frac{9 \cdot 2}{4}$

Step 16.2.8

Multiply $9$ by $2$ .

$f (\frac{3 π}{4}) = - \frac{18}{4}$

Step 16.2.9

Cancel the common factor of $18$ and $4$ .

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Step 16.2.9.1

Factor $2$ out of $18$ .

$f (\frac{3 π}{4}) = - \frac{2 (9)}{4}$

Step 16.2.9.2

Cancel the common factors.

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Step 16.2.9.2.1

Factor $2$ out of $4$ .

$f (\frac{3 π}{4}) = - \frac{2 \cdot 9}{2 \cdot 2}$

Step 16.2.9.2.2

Cancel the common factor.

$f (\frac{3 π}{4}) = - \frac{2 \cdot 9}{2 \cdot 2}$

Step 16.2.9.2.3

Rewrite the expression.

$f (\frac{3 π}{4}) = - \frac{9}{2}$

Step 16.2.10

The final answer is $- \frac{9}{2}$ .

$y = - \frac{9}{2}$

Step 17

Evaluate the second derivative at $x = \frac{π}{4}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 36 \cos (\frac{π}{4}) \sin (\frac{π}{4})$

Step 18

Evaluate the second derivative.

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Step 18.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- 36 \frac{\sqrt{2}}{2} \sin (\frac{π}{4})$

Step 18.2

Cancel the common factor of $2$ .

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Step 18.2.1

Factor $2$ out of $- 36$ .

$2 (- 18) \frac{\sqrt{2}}{2} \sin (\frac{π}{4})$

Step 18.2.2

Cancel the common factor.

$2 \cdot - 18 \frac{\sqrt{2}}{2} \sin (\frac{π}{4})$

Step 18.2.3

Rewrite the expression.

$- 18 \sqrt{2} \sin (\frac{π}{4})$

Step 18.3

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- 18 \sqrt{2} \frac{\sqrt{2}}{2}$

Step 18.4

Cancel the common factor of $2$ .

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Step 18.4.1

Factor $2$ out of $- 18 \sqrt{2}$ .

$2 (- 9 \sqrt{2}) \frac{\sqrt{2}}{2}$

Step 18.4.2

Cancel the common factor.

$2 (- 9 \sqrt{2}) \frac{\sqrt{2}}{2}$

Step 18.4.3

Rewrite the expression.

$- 9 \sqrt{2} \sqrt{2}$

Step 18.5

Raise $\sqrt{2}$ to the power of $1$ .

$- 9 ({\sqrt{2}}^{1} \sqrt{2})$

Step 18.6

Raise $\sqrt{2}$ to the power of $1$ .

$- 9 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

Step 18.7

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

$- 9 {\sqrt{2}}^{1 + 1}$

Step 18.8

Add $1$ and $1$ .

$- 9 {\sqrt{2}}^{2}$

Step 18.9

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

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Step 18.9.1

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

$- 9 {(2^{\frac{1}{2}})}^{2}$

Step 18.9.2

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

$- 9 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 18.9.3

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

$- 9 \cdot 2^{\frac{2}{2}}$

Step 18.9.4

Cancel the common factor of $2$ .

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Step 18.9.4.1

Cancel the common factor.

$- 9 \cdot 2^{\frac{2}{2}}$

Step 18.9.4.2

Rewrite the expression.

$- 9 \cdot 2^{1}$

Step 18.9.5

Evaluate the exponent.

$- 9 \cdot 2$

Step 18.10

Multiply $- 9$ by $2$ .

$- 18$

Step 19

$x = \frac{π}{4}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{π}{4}$ is a local maximum

Step 20

Find the y-value when $x = \frac{π}{4}$ .

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Step 20.1

Replace the variable $x$ with $\frac{π}{4}$ in the expression.

$f (\frac{π}{4}) = 9 \sin (\frac{π}{4}) \cos (\frac{π}{4})$

Step 20.2

Simplify the result.

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Step 20.2.1

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$f (\frac{π}{4}) = 9 (\frac{\sqrt{2}}{2}) \cdot \cos (\frac{π}{4})$

Step 20.2.2

Combine $9$ and $\frac{\sqrt{2}}{2}$ .

$f (\frac{π}{4}) = \frac{9 \sqrt{2}}{2} \cdot \cos (\frac{π}{4})$

Step 20.2.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$f (\frac{π}{4}) = \frac{9 \sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}$

Step 20.2.4

Multiply $\frac{9 \sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 20.2.4.1

Multiply $\frac{9 \sqrt{2}}{2}$ by $\frac{\sqrt{2}}{2}$ .

$f (\frac{π}{4}) = \frac{9 \sqrt{2} \sqrt{2}}{2 \cdot 2}$

Step 20.2.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$f (\frac{π}{4}) = \frac{9 (\sqrt{2} \sqrt{2})}{2 \cdot 2}$

Step 20.2.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$f (\frac{π}{4}) = \frac{9 (\sqrt{2} \sqrt{2})}{2 \cdot 2}$

Step 20.2.4.4

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

$f (\frac{π}{4}) = \frac{9 {\sqrt{2}}^{1 + 1}}{2 \cdot 2}$

Step 20.2.4.5

Add $1$ and $1$ .

$f (\frac{π}{4}) = \frac{9 {\sqrt{2}}^{2}}{2 \cdot 2}$

Step 20.2.4.6

Multiply $2$ by $2$ .

$f (\frac{π}{4}) = \frac{9 {\sqrt{2}}^{2}}{4}$

Step 20.2.5

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

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Step 20.2.5.1

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

$f (\frac{π}{4}) = \frac{9 {(2^{\frac{1}{2}})}^{2}}{4}$

Step 20.2.5.2

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

$f (\frac{π}{4}) = \frac{9 \cdot 2^{\frac{1}{2} \cdot 2}}{4}$

Step 20.2.5.3

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

$f (\frac{π}{4}) = \frac{9 \cdot 2^{\frac{2}{2}}}{4}$

Step 20.2.5.4

Cancel the common factor of $2$ .

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Step 20.2.5.4.1

Cancel the common factor.

$f (\frac{π}{4}) = \frac{9 \cdot 2^{\frac{2}{2}}}{4}$

Step 20.2.5.4.2

Rewrite the expression.

$f (\frac{π}{4}) = \frac{9 \cdot 2}{4}$

Step 20.2.5.5

Evaluate the exponent.

$f (\frac{π}{4}) = \frac{9 \cdot 2}{4}$

Step 20.2.6

Multiply $9$ by $2$ .

$f (\frac{π}{4}) = \frac{18}{4}$

Step 20.2.7

Cancel the common factor of $18$ and $4$ .

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Step 20.2.7.1

Factor $2$ out of $18$ .

$f (\frac{π}{4}) = \frac{2 (9)}{4}$

Step 20.2.7.2

Cancel the common factors.

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Step 20.2.7.2.1

Factor $2$ out of $4$ .

$f (\frac{π}{4}) = \frac{2 \cdot 9}{2 \cdot 2}$

Step 20.2.7.2.2

Cancel the common factor.

$f (\frac{π}{4}) = \frac{2 \cdot 9}{2 \cdot 2}$

Step 20.2.7.2.3

Rewrite the expression.

$f (\frac{π}{4}) = \frac{9}{2}$

Step 20.2.8

The final answer is $\frac{9}{2}$ .

$y = \frac{9}{2}$

Step 21

Evaluate the second derivative at $x = \frac{5 π}{4}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 36 \cos (\frac{5 π}{4}) \sin (\frac{5 π}{4})$

Step 22

Evaluate the second derivative.

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Step 22.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$- 36 (- \cos (\frac{π}{4})) \sin (\frac{5 π}{4})$

Step 22.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- 36 (- \frac{\sqrt{2}}{2}) \sin (\frac{5 π}{4})$

Step 22.3

Cancel the common factor of $2$ .

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Step 22.3.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

$- 36 \frac{- \sqrt{2}}{2} \sin (\frac{5 π}{4})$

Step 22.3.2

Factor $2$ out of $- 36$ .

$2 (- 18) \frac{- \sqrt{2}}{2} \sin (\frac{5 π}{4})$

Step 22.3.3

Cancel the common factor.

$2 \cdot - 18 \frac{- \sqrt{2}}{2} \sin (\frac{5 π}{4})$

Step 22.3.4

Rewrite the expression.

$- 18 (- \sqrt{2}) \sin (\frac{5 π}{4})$

Step 22.4

Multiply $- 1$ by $- 18$ .

$18 \sqrt{2} \sin (\frac{5 π}{4})$

Step 22.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.

$18 \sqrt{2} (- \sin (\frac{π}{4}))$

Step 22.6

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$18 \sqrt{2} (- \frac{\sqrt{2}}{2})$

Step 22.7

Cancel the common factor of $2$ .

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Step 22.7.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

$18 \sqrt{2} \frac{- \sqrt{2}}{2}$

Step 22.7.2

Factor $2$ out of $18 \sqrt{2}$ .

$2 (9 \sqrt{2}) \frac{- \sqrt{2}}{2}$

Step 22.7.3

Cancel the common factor.

$2 (9 \sqrt{2}) \frac{- \sqrt{2}}{2}$

Step 22.7.4

Rewrite the expression.

$9 \sqrt{2} (- \sqrt{2})$

Step 22.8

Multiply $- 1$ by $9$ .

$- 9 \sqrt{2} \sqrt{2}$

Step 22.9

Raise $\sqrt{2}$ to the power of $1$ .

$- 9 ({\sqrt{2}}^{1} \sqrt{2})$

Step 22.10

Raise $\sqrt{2}$ to the power of $1$ .

$- 9 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

Step 22.11

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

$- 9 {\sqrt{2}}^{1 + 1}$

Step 22.12

Add $1$ and $1$ .

$- 9 {\sqrt{2}}^{2}$

Step 22.13

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

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Step 22.13.1

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

$- 9 {(2^{\frac{1}{2}})}^{2}$

Step 22.13.2

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

$- 9 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 22.13.3

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

$- 9 \cdot 2^{\frac{2}{2}}$

Step 22.13.4

Cancel the common factor of $2$ .

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Step 22.13.4.1

Cancel the common factor.

$- 9 \cdot 2^{\frac{2}{2}}$

Step 22.13.4.2

Rewrite the expression.

$- 9 \cdot 2^{1}$

Step 22.13.5

Evaluate the exponent.

$- 9 \cdot 2$

Step 22.14

Multiply $- 9$ by $2$ .

$- 18$

Step 23

$x = \frac{5 π}{4}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{5 π}{4}$ is a local maximum

Step 24

Find the y-value when $x = \frac{5 π}{4}$ .

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Step 24.1

Replace the variable $x$ with $\frac{5 π}{4}$ in the expression.

$f (\frac{5 π}{4}) = 9 \sin (\frac{5 π}{4}) \cos (\frac{5 π}{4})$

Step 24.2

Simplify the result.

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Step 24.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.

$f (\frac{5 π}{4}) = 9 (- \sin (\frac{π}{4})) \cos (\frac{5 π}{4})$

Step 24.2.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$f (\frac{5 π}{4}) = 9 (- \frac{\sqrt{2}}{2}) \cdot \cos (\frac{5 π}{4})$

Step 24.2.3

Multiply $9 (- \frac{\sqrt{2}}{2})$ .

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Step 24.2.3.1

Multiply $- 1$ by $9$ .

$f (\frac{5 π}{4}) = - 9 \frac{\sqrt{2}}{2} \cdot \cos (\frac{5 π}{4})$

Step 24.2.3.2

Combine $- 9$ and $\frac{\sqrt{2}}{2}$ .

$f (\frac{5 π}{4}) = \frac{- 9 \sqrt{2}}{2} \cdot \cos (\frac{5 π}{4})$

Step 24.2.4

Move the negative in front of the fraction.

$f (\frac{5 π}{4}) = - \frac{9 \sqrt{2}}{2} \cdot \cos (\frac{5 π}{4})$

Step 24.2.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$f (\frac{5 π}{4}) = - \frac{9 \sqrt{2}}{2} \cdot (- \cos (\frac{π}{4}))$

Step 24.2.6

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$f (\frac{5 π}{4}) = - \frac{9 \sqrt{2}}{2} \cdot (- \frac{\sqrt{2}}{2})$

Step 24.2.7

Multiply $- \frac{9 \sqrt{2}}{2} (- \frac{\sqrt{2}}{2})$ .

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Step 24.2.7.1

Multiply $- 1$ by $- 1$ .

$f (\frac{5 π}{4}) = 1 (\frac{9 \sqrt{2}}{2}) \cdot \frac{\sqrt{2}}{2}$

Step 24.2.7.2

Multiply $\frac{9 \sqrt{2}}{2}$ by $1$ .

$f (\frac{5 π}{4}) = \frac{9 \sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}$

Step 24.2.7.3

Multiply $\frac{9 \sqrt{2}}{2}$ by $\frac{\sqrt{2}}{2}$ .

$f (\frac{5 π}{4}) = \frac{9 \sqrt{2} \sqrt{2}}{2 \cdot 2}$

Step 24.2.7.4

Raise $\sqrt{2}$ to the power of $1$ .

$f (\frac{5 π}{4}) = \frac{9 (\sqrt{2} \sqrt{2})}{2 \cdot 2}$

Step 24.2.7.5

Raise $\sqrt{2}$ to the power of $1$ .

$f (\frac{5 π}{4}) = \frac{9 (\sqrt{2} \sqrt{2})}{2 \cdot 2}$

Step 24.2.7.6

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

$f (\frac{5 π}{4}) = \frac{9 {\sqrt{2}}^{1 + 1}}{2 \cdot 2}$

Step 24.2.7.7

Add $1$ and $1$ .

$f (\frac{5 π}{4}) = \frac{9 {\sqrt{2}}^{2}}{2 \cdot 2}$

Step 24.2.7.8

Multiply $2$ by $2$ .

$f (\frac{5 π}{4}) = \frac{9 {\sqrt{2}}^{2}}{4}$

Step 24.2.8

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

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Step 24.2.8.1

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

$f (\frac{5 π}{4}) = \frac{9 {(2^{\frac{1}{2}})}^{2}}{4}$

Step 24.2.8.2

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

$f (\frac{5 π}{4}) = \frac{9 \cdot 2^{\frac{1}{2} \cdot 2}}{4}$

Step 24.2.8.3

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

$f (\frac{5 π}{4}) = \frac{9 \cdot 2^{\frac{2}{2}}}{4}$

Step 24.2.8.4

Cancel the common factor of $2$ .

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Step 24.2.8.4.1

Cancel the common factor.

$f (\frac{5 π}{4}) = \frac{9 \cdot 2^{\frac{2}{2}}}{4}$

Step 24.2.8.4.2

Rewrite the expression.

$f (\frac{5 π}{4}) = \frac{9 \cdot 2}{4}$

Step 24.2.8.5

Evaluate the exponent.

$f (\frac{5 π}{4}) = \frac{9 \cdot 2}{4}$

Step 24.2.9

Multiply $9$ by $2$ .

$f (\frac{5 π}{4}) = \frac{18}{4}$

Step 24.2.10

Cancel the common factor of $18$ and $4$ .

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Step 24.2.10.1

Factor $2$ out of $18$ .

$f (\frac{5 π}{4}) = \frac{2 (9)}{4}$

Step 24.2.10.2

Cancel the common factors.

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Step 24.2.10.2.1

Factor $2$ out of $4$ .

$f (\frac{5 π}{4}) = \frac{2 \cdot 9}{2 \cdot 2}$

Step 24.2.10.2.2

Cancel the common factor.

$f (\frac{5 π}{4}) = \frac{2 \cdot 9}{2 \cdot 2}$

Step 24.2.10.2.3

Rewrite the expression.

$f (\frac{5 π}{4}) = \frac{9}{2}$

Step 24.2.11

The final answer is $\frac{9}{2}$ .

$y = \frac{9}{2}$

Step 25

These are the local extrema for $f (x) = 9 \sin (x) \cos (x)$ .

$(- \frac{π}{4}, - \frac{9}{2})$ is a local minima

$(\frac{3 π}{4}, - \frac{9}{2})$ is a local minima

$(\frac{π}{4}, \frac{9}{2})$ is a local maxima

$(\frac{5 π}{4}, \frac{9}{2})$ is a local maxima

Step 26