Calculus Examples

$f (x) = \sin (x) \cdot \cos^{3} (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

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 $g (x) = \cos^{3} (x)$ .

$\sin (x) \frac{d}{d x} [\cos^{3} (x)] + \cos^{3} (x) \frac{d}{d x} [\sin (x)]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \cos (x)$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $\cos (x)$ .

$\sin (x) (\frac{d}{d u} [u^{3}] \frac{d}{d x} [\cos (x)]) + \cos^{3} (x) \frac{d}{d x} [\sin (x)]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$\sin (x) (3 u^{2} \frac{d}{d x} [\cos (x)]) + \cos^{3} (x) \frac{d}{d x} [\sin (x)]$

Step 1.2.3

Replace all occurrences of $u$ with $\cos (x)$ .

$\sin (x) (3 \cos^{2} (x) \frac{d}{d x} [\cos (x)]) + \cos^{3} (x) \frac{d}{d x} [\sin (x)]$

Step 1.3

Move $3$ to the left of $\sin (x)$ .

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

Step 1.4

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

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

Step 1.5

Multiply $- 1$ by $3$ .

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

Add $1$ and $1$ .

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

Step 1.10

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

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

Step 1.11

Multiply $\cos^{3} (x)$ by $\cos (x)$ by adding the exponents.

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Step 1.11.1

Multiply $\cos^{3} (x)$ by $\cos (x)$ .

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Step 1.11.1.1

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

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

Step 1.11.1.2

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

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

Step 1.11.2

Add $3$ and $1$ .

$- 3 \sin^{2} (x) \cos^{2} (x) + \cos^{4} (x)$

Step 1.12

Reorder terms.

$- 3 \cos^{2} (x) \sin^{2} (x) + \cos^{4} (x)$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- 3 \cos^{2} (x) \sin^{2} (x) + \cos^{4} (x)$ with respect to $x$ is $\frac{d}{d x} [- 3 \cos^{2} (x) \sin^{2} (x)] + \frac{d}{d x} [\cos^{4} (x)]$ .

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

Step 2.2

Evaluate $\frac{d}{d x} [- 3 \cos^{2} (x) \sin^{2} (x)]$ .

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Step 2.2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 \cos^{2} (x) \sin^{2} (x)$ with respect to $x$ is $- 3 \frac{d}{d x} [\cos^{2} (x) \sin^{2} (x)]$ .

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

Step 2.2.2

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 $f (x) = \cos^{2} (x)$ and $g (x) = \sin^{2} (x)$ .

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

Step 2.2.3

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 2.2.3.1

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

$f'' (x) = - 3 (\cos^{2} (x) (\frac{d}{d u (1)} ({u_{1}}^{2}) \frac{d}{d x} (\sin (x))) + \sin^{2} (x) \frac{d}{d x} (\cos^{2} (x))) + \frac{d}{d x} (\cos^{4} (x))$

Step 2.2.3.2

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

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

Step 2.2.3.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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Step 2.2.5.1

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

$f'' (x) = - 3 (\cos^{2} (x) (2 \sin (x) \cos (x)) + \sin^{2} (x) (\frac{d}{d u (2)} ({u_{2}}^{2}) \frac{d}{d x} (\cos (x)))) + \frac{d}{d x} (\cos^{4} (x))$

Step 2.2.5.2

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

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

Step 2.2.5.3

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

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

Step 2.2.6

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

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

Step 2.2.7

Multiply $\cos^{2} (x)$ by $\cos (x)$ by adding the exponents.

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Step 2.2.7.1

Move $\cos (x)$ .

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

Step 2.2.7.2

Multiply $\cos (x)$ by $\cos^{2} (x)$ .

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Step 2.2.7.2.1

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

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

Step 2.2.7.2.2

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

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

Step 2.2.7.3

Add $1$ and $2$ .

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

Step 2.2.8

Move $2$ to the left of $\cos^{3} (x)$ .

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

Step 2.2.9

Multiply $- 1$ by $2$ .

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

Step 2.2.10

Multiply $\sin^{2} (x)$ by $\sin (x)$ by adding the exponents.

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Step 2.2.10.1

Move $\sin (x)$ .

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

Step 2.2.10.2

Multiply $\sin (x)$ by $\sin^{2} (x)$ .

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Step 2.2.10.2.1

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

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

Step 2.2.10.2.2

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

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

Step 2.2.10.3

Add $1$ and $2$ .

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

Step 2.3

Evaluate $\frac{d}{d x} [\cos^{4} (x)]$ .

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Step 2.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = \cos (x)$ .

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Step 2.3.1.1

To apply the Chain Rule, set $u_{3}$ as $\cos (x)$ .

$f'' (x) = - 3 (2 \cos^{3} (x) \sin (x) + \sin^{3} (x) (- 2 \cos (x))) + \frac{d}{d u (3)} ({u_{3}}^{4}) \frac{d}{d x} (\cos (x))$

Step 2.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 4$ .

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

Step 2.3.1.3

Replace all occurrences of $u_{3}$ with $\cos (x)$ .

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $- 1$ by $4$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$f'' (x) = - 3 (2 \cos^{3} (x) \sin (x)) - 3 (\sin^{3} (x) (- 2 \cos (x))) - 4 \cos^{3} (x) \sin (x)$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $2$ by $- 3$ .

$f'' (x) = - 6 (\cos^{3} (x) \sin (x)) - 3 (\sin^{3} (x) (- 2 \cos (x))) - 4 \cos^{3} (x) \sin (x)$

Step 2.4.2.2

Multiply $- 2$ by $- 3$ .

$f'' (x) = - 6 \cos^{3} (x) \sin (x) + 6 (\sin^{3} (x) (\cos (x))) - 4 \cos^{3} (x) \sin (x)$

Step 2.4.2.3

Subtract $4 \cos^{3} (x) \sin (x)$ from $- 6 \cos^{3} (x) \sin (x)$ .

$f'' (x) = - 10 \cos^{3} (x) \sin (x) + 6 \sin^{3} (x) \cos (x)$

Step 3

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

$- 3 \cos^{2} (x) \sin^{2} (x) + \cos^{4} (x) = 0$

Step 4

Factor $\cos^{2} (x)$ out of $- 3 \cos^{2} (x) \sin^{2} (x) + \cos^{4} (x)$ .

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Step 4.1

Factor $\cos^{2} (x)$ out of $- 3 \cos^{2} (x) \sin^{2} (x)$ .

$\cos^{2} (x) (- 3 \sin^{2} (x)) + \cos^{4} (x) = 0$

Step 4.2

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

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

Step 4.3

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

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

Step 5

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

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

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

Step 6

Set $\cos^{2} (x)$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $\cos^{2} (x)$ equal to $0$ .

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

Step 6.2

Solve $\cos^{2} (x) = 0$ for $x$ .

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Step 6.2.1

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

$\cos (x) = \pm \sqrt{0}$

Step 6.2.2

Simplify $\pm \sqrt{0}$ .

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Step 6.2.2.1

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

$\cos (x) = \pm \sqrt{0^{2}}$

Step 6.2.2.2

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

$\cos (x) = \pm 0$

Step 6.2.2.3

Plus or minus $0$ is $0$ .

$\cos (x) = 0$

Step 6.2.3

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

$x = \arccos (0)$

Step 6.2.4

Simplify the right side.

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Step 6.2.4.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 6.2.5

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{2}$

Step 6.2.6

Simplify $2 π - \frac{π}{2}$ .

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Step 6.2.6.1

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

$x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 6.2.6.2

Combine fractions.

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Step 6.2.6.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 6.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 6.2.6.3

Simplify the numerator.

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Step 6.2.6.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 6.2.6.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 6.2.7

The solution to the equation $x = \frac{π}{2}$ .

$x = \frac{π}{2}, \frac{3 π}{2}$

Step 7

Set $- 3 \sin^{2} (x) + \cos^{2} (x)$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $- 3 \sin^{2} (x) + \cos^{2} (x)$ equal to $0$ .

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

Step 7.2

Solve $- 3 \sin^{2} (x) + \cos^{2} (x) = 0$ for $x$ .

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Step 7.2.1

Replace the $- 3 \sin^{2} (x)$ with $- 3 (1 - \cos^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

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

Step 7.2.2

Simplify each term.

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Step 7.2.2.1

Apply the distributive property.

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

Step 7.2.2.2

Multiply $- 3$ by $1$ .

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

Step 7.2.2.3

Multiply $- 1$ by $- 3$ .

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

Step 7.2.3

Add $3 \cos^{2} (x)$ and $\cos^{2} (x)$ .

$- 3 + 4 \cos^{2} (x) = 0$

Step 7.2.4

Reorder the polynomial.

$4 \cos^{2} (x) - 3 = 0$

Step 7.2.5

Add $3$ to both sides of the equation.

$4 \cos^{2} (x) = 3$

Step 7.2.6

Divide each term in $4 \cos^{2} (x) = 3$ by $4$ and simplify.

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Step 7.2.6.1

Divide each term in $4 \cos^{2} (x) = 3$ by $4$ .

$\frac{4 \cos^{2} (x)}{4} = \frac{3}{4}$

Step 7.2.6.2

Simplify the left side.

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Step 7.2.6.2.1

Cancel the common factor of $4$ .

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Step 7.2.6.2.1.1

Cancel the common factor.

$\frac{4 \cos^{2} (x)}{4} = \frac{3}{4}$

Step 7.2.6.2.1.2

Divide $\cos^{2} (x)$ by $1$ .

$\cos^{2} (x) = \frac{3}{4}$

Step 7.2.7

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

$\cos (x) = \pm \sqrt{\frac{3}{4}}$

Step 7.2.8

Simplify $\pm \sqrt{\frac{3}{4}}$ .

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Step 7.2.8.1

Rewrite $\sqrt{\frac{3}{4}}$ as $\frac{\sqrt{3}}{\sqrt{4}}$ .

$\cos (x) = \pm \frac{\sqrt{3}}{\sqrt{4}}$

Step 7.2.8.2

Simplify the denominator.

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Step 7.2.8.2.1

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

$\cos (x) = \pm \frac{\sqrt{3}}{\sqrt{2^{2}}}$

Step 7.2.8.2.2

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

$\cos (x) = \pm \frac{\sqrt{3}}{2}$

Step 7.2.9

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

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Step 7.2.9.1

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

$\cos (x) = \frac{\sqrt{3}}{2}$

Step 7.2.9.2

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

$\cos (x) = - \frac{\sqrt{3}}{2}$

Step 7.2.9.3

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

$\cos (x) = \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}$

Step 7.2.10

Set up each of the solutions to solve for $x$ .

$\cos (x) = \frac{\sqrt{3}}{2}$

$\cos (x) = - \frac{\sqrt{3}}{2}$

Step 7.2.11

Solve for $x$ in $\cos (x) = \frac{\sqrt{3}}{2}$ .

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Step 7.2.11.1

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

$x = \arccos (\frac{\sqrt{3}}{2})$

Step 7.2.11.2

Simplify the right side.

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Step 7.2.11.2.1

The exact value of $\arccos (\frac{\sqrt{3}}{2})$ is $\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 7.2.11.3

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{6}$

Step 7.2.11.4

Simplify $2 π - \frac{π}{6}$ .

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Step 7.2.11.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = 2 π \cdot \frac{6}{6} - \frac{π}{6}$

Step 7.2.11.4.2

Combine fractions.

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Step 7.2.11.4.2.1

Combine $2 π$ and $\frac{6}{6}$ .

$x = \frac{2 π \cdot 6}{6} - \frac{π}{6}$

Step 7.2.11.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - π}{6}$

Step 7.2.11.4.3

Simplify the numerator.

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Step 7.2.11.4.3.1

Multiply $6$ by $2$ .

$x = \frac{12 π - π}{6}$

Step 7.2.11.4.3.2

Subtract $π$ from $12 π$ .

$x = \frac{11 π}{6}$

Step 7.2.11.5

The solution to the equation $x = \frac{π}{6}$ .

$x = \frac{π}{6}, \frac{11 π}{6}$

Step 7.2.12

Solve for $x$ in $\cos (x) = - \frac{\sqrt{3}}{2}$ .

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Step 7.2.12.1

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

$x = \arccos (- \frac{\sqrt{3}}{2})$

Step 7.2.12.2

Simplify the right side.

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Step 7.2.12.2.1

The exact value of $\arccos (- \frac{\sqrt{3}}{2})$ is $\frac{5 π}{6}$ .

$x = \frac{5 π}{6}$

Step 7.2.12.3

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - \frac{5 π}{6}$

Step 7.2.12.4

Simplify $2 π - \frac{5 π}{6}$ .

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Step 7.2.12.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = 2 π \cdot \frac{6}{6} - \frac{5 π}{6}$

Step 7.2.12.4.2

Combine fractions.

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Step 7.2.12.4.2.1

Combine $2 π$ and $\frac{6}{6}$ .

$x = \frac{2 π \cdot 6}{6} - \frac{5 π}{6}$

Step 7.2.12.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - 5 π}{6}$

Step 7.2.12.4.3

Simplify the numerator.

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Step 7.2.12.4.3.1

Multiply $6$ by $2$ .

$x = \frac{12 π - 5 π}{6}$

Step 7.2.12.4.3.2

Subtract $5 π$ from $12 π$ .

$x = \frac{7 π}{6}$

Step 7.2.12.5

The solution to the equation $x = \frac{5 π}{6}$ .

$x = \frac{5 π}{6}, \frac{7 π}{6}$

Step 7.2.13

List all of the solutions.

$x = \frac{π}{6}, \frac{11 π}{6}, \frac{5 π}{6}, \frac{7 π}{6}$

Step 8

The final solution is all the values that make $\cos^{2} (x) (- 3 \sin^{2} (x) + \cos^{2} (x)) = 0$ true.

$x = \frac{π}{2}, \frac{3 π}{2}, \frac{π}{6}, \frac{11 π}{6}, \frac{5 π}{6}, \frac{7 π}{6}$

Step 9

Evaluate the second derivative at $x = \frac{π}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 10 \cos^{3} (\frac{π}{2}) \sin (\frac{π}{2}) + 6 \sin^{3} (\frac{π}{2}) \cos (\frac{π}{2})$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify each term.

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Step 10.1.1

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$- 10 \cdot 0^{3} \sin (\frac{π}{2}) + 6 \sin^{3} (\frac{π}{2}) \cos (\frac{π}{2})$

Step 10.1.2

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

$- 10 \cdot 0 \sin (\frac{π}{2}) + 6 \sin^{3} (\frac{π}{2}) \cos (\frac{π}{2})$

Step 10.1.3

Multiply $- 10$ by $0$ .

$0 \sin (\frac{π}{2}) + 6 \sin^{3} (\frac{π}{2}) \cos (\frac{π}{2})$

Step 10.1.4

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$0 \cdot 1 + 6 \sin^{3} (\frac{π}{2}) \cos (\frac{π}{2})$

Step 10.1.5

Multiply $0$ by $1$ .

$0 + 6 \sin^{3} (\frac{π}{2}) \cos (\frac{π}{2})$

Step 10.1.6

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$0 + 6 \cdot 1^{3} \cos (\frac{π}{2})$

Step 10.1.7

One to any power is one.

$0 + 6 \cdot 1 \cos (\frac{π}{2})$

Step 10.1.8

Multiply $6$ by $1$ .

$0 + 6 \cos (\frac{π}{2})$

Step 10.1.9

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$0 + 6 \cdot 0$

Step 10.1.10

Multiply $6$ by $0$ .

$0 + 0$

Step 10.2

Add $0$ and $0$ .

$0$

Step 11

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 11.1

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

$(- \infty, \frac{π}{6}) \cup (\frac{π}{6}, \frac{π}{2}) \cup (\frac{π}{2}, \frac{5 π}{6}) \cup (\frac{5 π}{6}, \frac{7 π}{6}) \cup (\frac{7 π}{6}, \frac{3 π}{2}) \cup (\frac{3 π}{2}, \frac{11 π}{6}) \cup (\frac{11 π}{6}, \infty)$

Step 11.2

Substitute any number, such as $0$ , from the interval $(- \infty, \frac{π}{6})$ in the first derivative $- 3 \cos^{2} (x) \sin^{2} (x) + \cos^{4} (x)$ to check if the result is negative or positive.

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Step 11.2.1

Replace the variable $x$ with $0$ in the expression.

$f' (0) = - 3 \cos^{2} (0) \sin^{2} (0) + \cos^{4} (0)$

Step 11.2.2

Simplify the result.

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Step 11.2.2.1

Simplify each term.

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Step 11.2.2.1.1

The exact value of $\cos (0)$ is $1$ .

$f' (0) = - 3 \cdot (1^{2} \sin^{2} (0)) + \cos^{4} (0)$

Step 11.2.2.1.2

One to any power is one.

$f' (0) = - 3 \cdot (1 \sin^{2} (0)) + \cos^{4} (0)$

Step 11.2.2.1.3

Multiply $- 3$ by $1$ .

$f' (0) = - 3 \sin^{2} (0) + \cos^{4} (0)$

Step 11.2.2.1.4

The exact value of $\sin (0)$ is $0$ .

$f' (0) = - 3 \cdot 0^{2} + \cos^{4} (0)$

Step 11.2.2.1.5

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

$f' (0) = - 3 \cdot 0 + \cos^{4} (0)$

Step 11.2.2.1.6

Multiply $- 3$ by $0$ .

$f' (0) = 0 + \cos^{4} (0)$

Step 11.2.2.1.7

The exact value of $\cos (0)$ is $1$ .

$f' (0) = 0 + 1^{4}$

Step 11.2.2.1.8

One to any power is one.

$f' (0) = 0 + 1$

Step 11.2.2.2

Add $0$ and $1$ .

$f' (0) = 1$

Step 11.2.2.3

The final answer is $1$ .

$1$

Step 11.3

Substitute any number, such as $1$ , from the interval $(\frac{π}{6}, \frac{π}{2})$ in the first derivative $- 3 \cos^{2} (x) \sin^{2} (x) + \cos^{4} (x)$ to check if the result is negative or positive.

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Step 11.3.1

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

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

Step 11.3.2

Simplify the result.

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Step 11.3.2.1

Simplify each term.

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Step 11.3.2.1.1

Evaluate $\cos (1)$ .

$f' (1) = - 3 \cdot ({0.5403023}^{2} \sin^{2} (1)) + \cos^{4} (1)$

Step 11.3.2.1.2

Raise $0.5403023$ to the power of $2$ .

$f' (1) = - 3 \cdot (0.29192658 \sin^{2} (1)) + \cos^{4} (1)$

Step 11.3.2.1.3

Multiply $- 3$ by $0.29192658$ .

$f' (1) = - 0.87577974 \sin^{2} (1) + \cos^{4} (1)$

Step 11.3.2.1.4

Evaluate $\sin (1)$ .

$f' (1) = - 0.87577974 \cdot {0.84147098}^{2} + \cos^{4} (1)$

Step 11.3.2.1.5

Raise $0.84147098$ to the power of $2$ .

$f' (1) = - 0.87577974 \cdot 0.70807341 + \cos^{4} (1)$

Step 11.3.2.1.6

Multiply $- 0.87577974$ by $0.70807341$ .

$f' (1) = - 0.62011635 + \cos^{4} (1)$

Step 11.3.2.1.7

Evaluate $\cos (1)$ .

$f' (1) = - 0.62011635 + {0.5403023}^{4}$

Step 11.3.2.1.8

Raise $0.5403023$ to the power of $4$ .

$f' (1) = - 0.62011635 + 0.08522112$

Step 11.3.2.2

Add $- 0.62011635$ and $0.08522112$ .

$f' (1) = - 0.53489522$

Step 11.3.2.3

The final answer is $- 0.53489522$ .

$- 0.53489522$

Step 11.4

Substitute any number, such as $2$ , from the interval $(\frac{π}{2}, \frac{5 π}{6})$ in the first derivative $- 3 \cos^{2} (x) \sin^{2} (x) + \cos^{4} (x)$ to check if the result is negative or positive.

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Step 11.4.1

Replace the variable $x$ with $2$ in the expression.

$f' (2) = - 3 \cos^{2} (2) \sin^{2} (2) + \cos^{4} (2)$

Step 11.4.2

Simplify the result.

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Step 11.4.2.1

Simplify each term.

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Step 11.4.2.1.1

Evaluate $\cos (2)$ .

$f' (2) = - 3 {(- 0.41614683)}^{2} \sin^{2} (2) + \cos^{4} (2)$

Step 11.4.2.1.2

Raise $- 0.41614683$ to the power of $2$ .

$f' (2) = - 3 \cdot (0.17317818 \sin^{2} (2)) + \cos^{4} (2)$

Step 11.4.2.1.3

Multiply $- 3$ by $0.17317818$ .

$f' (2) = - 0.51953456 \sin^{2} (2) + \cos^{4} (2)$

Step 11.4.2.1.4

Evaluate $\sin (2)$ .

$f' (2) = - 0.51953456 \cdot {0.90929742}^{2} + \cos^{4} (2)$

Step 11.4.2.1.5

Raise $0.90929742$ to the power of $2$ .

$f' (2) = - 0.51953456 \cdot 0.82682181 + \cos^{4} (2)$

Step 11.4.2.1.6

Multiply $- 0.51953456$ by $0.82682181$ .

$f' (2) = - 0.42956251 + \cos^{4} (2)$

Step 11.4.2.1.7

Evaluate $\cos (2)$ .

$f' (2) = - 0.42956251 + {(- 0.41614683)}^{4}$

Step 11.4.2.1.8

Raise $- 0.41614683$ to the power of $4$ .

$f' (2) = - 0.42956251 + 0.02999068$

Step 11.4.2.2

Add $- 0.42956251$ and $0.02999068$ .

$f' (2) = - 0.39957182$

Step 11.4.2.3

The final answer is $- 0.39957182$ .

$- 0.39957182$

Step 11.5

Substitute any number, such as $3$ , from the interval $(\frac{5 π}{6}, \frac{7 π}{6})$ in the first derivative $- 3 \cos^{2} (x) \sin^{2} (x) + \cos^{4} (x)$ to check if the result is negative or positive.

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Step 11.5.1

Replace the variable $x$ with $3$ in the expression.

$f' (3) = - 3 \cos^{2} (3) \sin^{2} (3) + \cos^{4} (3)$

Step 11.5.2

Simplify the result.

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Step 11.5.2.1

Simplify each term.

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Step 11.5.2.1.1

Evaluate $\cos (3)$ .

$f' (3) = - 3 {(- 0.98999249)}^{2} \sin^{2} (3) + \cos^{4} (3)$

Step 11.5.2.1.2

Raise $- 0.98999249$ to the power of $2$ .

$f' (3) = - 3 \cdot (0.98008514 \sin^{2} (3)) + \cos^{4} (3)$

Step 11.5.2.1.3

Multiply $- 3$ by $0.98008514$ .

$f' (3) = - 2.94025542 \sin^{2} (3) + \cos^{4} (3)$

Step 11.5.2.1.4

Evaluate $\sin (3)$ .

$f' (3) = - 2.94025542 \cdot {0.14112}^{2} + \cos^{4} (3)$

Step 11.5.2.1.5

Raise $0.14112$ to the power of $2$ .

$f' (3) = - 2.94025542 \cdot 0.01991485 + \cos^{4} (3)$

Step 11.5.2.1.6

Multiply $- 2.94025542$ by $0.01991485$ .

$f' (3) = - 0.05855476 + \cos^{4} (3)$

Step 11.5.2.1.7

Evaluate $\cos (3)$ .

$f' (3) = - 0.05855476 + {(- 0.98999249)}^{4}$

Step 11.5.2.1.8

Raise $- 0.98999249$ to the power of $4$ .

$f' (3) = - 0.05855476 + 0.96056688$

Step 11.5.2.2

Add $- 0.05855476$ and $0.96056688$ .

$f' (3) = 0.90201212$

Step 11.5.2.3

The final answer is $0.90201212$ .

$0.90201212$

Step 11.6

Substitute any number, such as $4$ , from the interval $(\frac{7 π}{6}, \frac{3 π}{2})$ in the first derivative $- 3 \cos^{2} (x) \sin^{2} (x) + \cos^{4} (x)$ to check if the result is negative or positive.

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Step 11.6.1

Replace the variable $x$ with $4$ in the expression.

$f' (4) = - 3 \cos^{2} (4) \sin^{2} (4) + \cos^{4} (4)$

Step 11.6.2

Simplify the result.

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Step 11.6.2.1

Simplify each term.

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Step 11.6.2.1.1

Evaluate $\cos (4)$ .

$f' (4) = - 3 {(- 0.65364362)}^{2} \sin^{2} (4) + \cos^{4} (4)$

Step 11.6.2.1.2

Raise $- 0.65364362$ to the power of $2$ .

$f' (4) = - 3 \cdot (0.42724998 \sin^{2} (4)) + \cos^{4} (4)$

Step 11.6.2.1.3

Multiply $- 3$ by $0.42724998$ .

$f' (4) = - 1.28174994 \sin^{2} (4) + \cos^{4} (4)$

Step 11.6.2.1.4

Evaluate $\sin (4)$ .

$f' (4) = - 1.28174994 {(- 0.75680249)}^{2} + \cos^{4} (4)$

Step 11.6.2.1.5

Raise $- 0.75680249$ to the power of $2$ .

$f' (4) = - 1.28174994 \cdot 0.57275001 + \cos^{4} (4)$

Step 11.6.2.1.6

Multiply $- 1.28174994$ by $0.57275001$ .

$f' (4) = - 0.7341223 + \cos^{4} (4)$

Step 11.6.2.1.7

Evaluate $\cos (4)$ .

$f' (4) = - 0.7341223 + {(- 0.65364362)}^{4}$

Step 11.6.2.1.8

Raise $- 0.65364362$ to the power of $4$ .

$f' (4) = - 0.7341223 + 0.18254254$

Step 11.6.2.2

Add $- 0.7341223$ and $0.18254254$ .

$f' (4) = - 0.55157975$

Step 11.6.2.3

The final answer is $- 0.55157975$ .

$- 0.55157975$

Step 11.7

Substitute any number, such as $5$ , from the interval $(\frac{3 π}{2}, \frac{11 π}{6})$ in the first derivative $- 3 \cos^{2} (x) \sin^{2} (x) + \cos^{4} (x)$ to check if the result is negative or positive.

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Step 11.7.1

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

$f' (5) = - 3 \cos^{2} (5) \sin^{2} (5) + \cos^{4} (5)$

Step 11.7.2

Simplify the result.

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Step 11.7.2.1

Simplify each term.

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Step 11.7.2.1.1

Evaluate $\cos (5)$ .

$f' (5) = - 3 \cdot ({0.28366218}^{2} \sin^{2} (5)) + \cos^{4} (5)$

Step 11.7.2.1.2

Raise $0.28366218$ to the power of $2$ .

$f' (5) = - 3 \cdot (0.08046423 \sin^{2} (5)) + \cos^{4} (5)$

Step 11.7.2.1.3

Multiply $- 3$ by $0.08046423$ .

$f' (5) = - 0.2413927 \sin^{2} (5) + \cos^{4} (5)$

Step 11.7.2.1.4

Evaluate $\sin (5)$ .

$f' (5) = - 0.2413927 {(- 0.95892427)}^{2} + \cos^{4} (5)$

Step 11.7.2.1.5

Raise $- 0.95892427$ to the power of $2$ .

$f' (5) = - 0.2413927 \cdot 0.91953576 + \cos^{4} (5)$

Step 11.7.2.1.6

Multiply $- 0.2413927$ by $0.91953576$ .

$f' (5) = - 0.22196922 + \cos^{4} (5)$

Step 11.7.2.1.7

Evaluate $\cos (5)$ .

$f' (5) = - 0.22196922 + {0.28366218}^{4}$

Step 11.7.2.1.8

Raise $0.28366218$ to the power of $4$ .

$f' (5) = - 0.22196922 + 0.00647449$

Step 11.7.2.2

Add $- 0.22196922$ and $0.00647449$ .

$f' (5) = - 0.21549473$

Step 11.7.2.3

The final answer is $- 0.21549473$ .

$- 0.21549473$

Step 11.8

Substitute any number, such as $8$ , from the interval $(\frac{11 π}{6}, \infty)$ in the first derivative $- 3 \cos^{2} (x) \sin^{2} (x) + \cos^{4} (x)$ to check if the result is negative or positive.

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Step 11.8.1

Replace the variable $x$ with $8$ in the expression.

$f' (8) = - 3 \cos^{2} (8) \sin^{2} (8) + \cos^{4} (8)$

Step 11.8.2

Simplify the result.

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Step 11.8.2.1

Simplify each term.

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Step 11.8.2.1.1

Evaluate $\cos (8)$ .

$f' (8) = - 3 {(- 0.14550003)}^{2} \sin^{2} (8) + \cos^{4} (8)$

Step 11.8.2.1.2

Raise $- 0.14550003$ to the power of $2$ .

$f' (8) = - 3 \cdot (0.02117025 \sin^{2} (8)) + \cos^{4} (8)$

Step 11.8.2.1.3

Multiply $- 3$ by $0.02117025$ .

$f' (8) = - 0.06351077 \sin^{2} (8) + \cos^{4} (8)$

Step 11.8.2.1.4

Evaluate $\sin (8)$ .

$f' (8) = - 0.06351077 \cdot {0.98935824}^{2} + \cos^{4} (8)$

Step 11.8.2.1.5

Raise $0.98935824$ to the power of $2$ .

$f' (8) = - 0.06351077 \cdot 0.97882974 + \cos^{4} (8)$

Step 11.8.2.1.6

Multiply $- 0.06351077$ by $0.97882974$ .

$f' (8) = - 0.06216623 + \cos^{4} (8)$

Step 11.8.2.1.7

Evaluate $\cos (8)$ .

$f' (8) = - 0.06216623 + {(- 0.14550003)}^{4}$

Step 11.8.2.1.8

Raise $- 0.14550003$ to the power of $4$ .

$f' (8) = - 0.06216623 + 0.00044817$

Step 11.8.2.2

Add $- 0.06216623$ and $0.00044817$ .

$f' (8) = - 0.06171805$

Step 11.8.2.3

The final answer is $- 0.06171805$ .

$- 0.06171805$

Step 11.9

Since the first derivative changed signs from positive to negative around $x = \frac{π}{6}$ , then $x = \frac{π}{6}$ is a local maximum.

$x = \frac{π}{6}$ is a local maximum

Step 11.10

Since the first derivative did not change signs around $x = \frac{π}{2}$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 11.11

Since the first derivative changed signs from negative to positive around $x = \frac{5 π}{6}$ , then $x = \frac{5 π}{6}$ is a local minimum.

$x = \frac{5 π}{6}$ is a local minimum

Step 11.12

Since the first derivative changed signs from positive to negative around $x = \frac{7 π}{6}$ , then $x = \frac{7 π}{6}$ is a local maximum.

$x = \frac{7 π}{6}$ is a local maximum

Step 11.13

Since the first derivative did not change signs around $x = \frac{3 π}{2}$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 11.14

These are the local extrema for $f (x) = \sin (x) \cdot \cos^{3} (x)$ .

$x = \frac{π}{6}$ is a local maximum

$x = \frac{5 π}{6}$ is a local minimum

$x = \frac{7 π}{6}$ is a local maximum

$x = \frac{π}{6}$ is a local maximum

$x = \frac{5 π}{6}$ is a local minimum

$x = \frac{7 π}{6}$ is a local maximum

Step 12