Calculus Examples

$f (x) = 8 \cos^{4} (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

Since $8$ is constant with respect to $x$ , the derivative of $8 \cos^{4} (x)$ with respect to $x$ is $8 \frac{d}{d x} [\cos^{4} (x)]$ .

$8 \frac{d}{d x} [\cos^{4} (x)]$

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 $f (x) = x^{4}$ and $g (x) = \cos (x)$ .

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Step 1.2.1

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

$8 (\frac{d}{d u} [u^{4}] \frac{d}{d x} [\cos (x)])$

Step 1.2.2

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

$8 (4 u^{3} \frac{d}{d x} [\cos (x)])$

Step 1.2.3

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

$8 (4 \cos^{3} (x) \frac{d}{d x} [\cos (x)])$

Step 1.3

Multiply $4$ by $8$ .

$32 (\cos^{3} (x) \frac{d}{d x} [\cos (x)])$

Step 1.4

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

$32 \cos^{3} (x) (- \sin (x))$

Step 1.5

Multiply $- 1$ by $32$ .

$- 32 \cos^{3} (x) \sin (x)$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- 32$ is constant with respect to $x$ , the derivative of $- 32 \cos^{3} (x) \sin (x)$ with respect to $x$ is $- 32 \frac{d}{d x} [\cos^{3} (x) \sin (x)]$ .

$f'' (x) = - 32 \frac{d}{d x} (\cos^{3} (x) \sin (x))$

Step 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 $f (x) = \cos^{3} (x)$ and $g (x) = \sin (x)$ .

$f'' (x) = - 32 (\cos^{3} (x) \frac{d}{d x} (\sin (x)) + \sin (x) \frac{d}{d x} (\cos^{3} (x)))$

Step 2.3

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

$f'' (x) = - 32 (\cos^{3} (x) \cos (x) + \sin (x) \frac{d}{d x} (\cos^{3} (x)))$

Step 2.4

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

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Step 2.4.1

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

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Step 2.4.1.1

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

$f'' (x) = - 32 (\cos^{3} (x) \cos (x) + \sin (x) \frac{d}{d x} (\cos^{3} (x)))$

Step 2.4.1.2

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

$f'' (x) = - 32 ({\cos (x)}^{3 + 1} + \sin (x) \frac{d}{d x} (\cos^{3} (x)))$

Step 2.4.2

Add $3$ and $1$ .

$f'' (x) = - 32 (\cos^{4} (x) + \sin (x) \frac{d}{d x} (\cos^{3} (x)))$

Step 2.5

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

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Step 2.5.1

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

$f'' (x) = - 32 (\cos^{4} (x) + \sin (x) (\frac{d}{d u} (u^{3}) \frac{d}{d x} (\cos (x))))$

Step 2.5.2

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

$f'' (x) = - 32 (\cos^{4} (x) + \sin (x) (3 u^{2} \frac{d}{d x} (\cos (x))))$

Step 2.5.3

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

$f'' (x) = - 32 (\cos^{4} (x) + \sin (x) (3 \cos^{2} (x) \frac{d}{d x} (\cos (x))))$

Step 2.6

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

$f'' (x) = - 32 (\cos^{4} (x) + 3 \cdot (\sin (x) \cos^{2} (x)) \frac{d}{d x} (\cos (x)))$

Step 2.7

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

$f'' (x) = - 32 (\cos^{4} (x) + 3 \sin (x) \cos^{2} (x) (- \sin (x)))$

Step 2.8

Multiply $- 1$ by $3$ .

$f'' (x) = - 32 (\cos^{4} (x) - 3 \sin (x) \cos^{2} (x) \sin (x))$

Step 2.9

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

$f'' (x) = - 32 (\cos^{4} (x) - 3 (\sin (x) \sin (x)) \cos^{2} (x))$

Step 2.10

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

$f'' (x) = - 32 (\cos^{4} (x) - 3 (\sin (x) \sin (x)) \cos^{2} (x))$

Step 2.11

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

$f'' (x) = - 32 (\cos^{4} (x) - 3 {\sin (x)}^{1 + 1} \cos^{2} (x))$

Step 2.12

Add $1$ and $1$ .

$f'' (x) = - 32 (\cos^{4} (x) - 3 \sin^{2} (x) \cos^{2} (x))$

Step 2.13

Simplify.

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Step 2.13.1

Apply the distributive property.

$f'' (x) = - 32 \cos^{4} (x) - 32 (- 3 \sin^{2} (x) \cos^{2} (x))$

Step 2.13.2

Multiply $- 3$ by $- 32$ .

$f'' (x) = - 32 \cos^{4} (x) + 96 \sin^{2} (x) \cos^{2} (x)$

Step 2.13.3

Reorder terms.

$f'' (x) = 96 \cos^{2} (x) \sin^{2} (x) - 32 \cos^{4} (x)$

Step 3

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

$- 32 \cos^{3} (x) \sin (x) = 0$

Step 4

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

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

$\sin (x) = 0$

Step 5

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

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Step 5.1

Set $\cos^{3} (x)$ equal to $0$ .

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

Step 5.2

Solve $\cos^{3} (x) = 0$ for $x$ .

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Step 5.2.1

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

$\cos (x) = \sqrt[3]{0}$

Step 5.2.2

Simplify $\sqrt[3]{0}$ .

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Step 5.2.2.1

Rewrite $0$ as $0^{3}$ .

$\cos (x) = \sqrt[3]{0^{3}}$

Step 5.2.2.2

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

$\cos (x) = 0$

Step 5.2.3

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

$x = \arccos (0)$

Step 5.2.4

Simplify the right side.

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Step 5.2.4.1

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

$x = \frac{π}{2}$

Step 5.2.5

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 5.2.6

Simplify $2 π - \frac{π}{2}$ .

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Step 5.2.6.1

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 5.2.6.2

Combine fractions.

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Step 5.2.6.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 5.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 5.2.6.3

Simplify the numerator.

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Step 5.2.6.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 5.2.6.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 5.2.7

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

$x = \frac{π}{2}, \frac{3 π}{2}$

Step 6

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

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Step 6.1

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

$\sin (x) = 0$

Step 6.2

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

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Step 6.2.1

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

$x = \arcsin (0)$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

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

$x = 0$

Step 6.2.3

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

$x = π - 0$

Step 6.2.4

Subtract $0$ from $π$ .

$x = π$

Step 6.2.5

The solution to the equation $x = 0$ .

$x = 0, π$

Step 7

The final solution is all the values that make $- 32 \cos^{3} (x) \sin (x) = 0$ true.

$x = \frac{π}{2}, \frac{3 π}{2}, 0, π$

Step 8

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.

$96 \cos^{2} (\frac{π}{2}) \sin^{2} (\frac{π}{2}) - 32 \cos^{4} (\frac{π}{2})$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

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

$96 \cdot 0^{2} \sin^{2} (\frac{π}{2}) - 32 \cos^{4} (\frac{π}{2})$

Step 9.1.2

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

$96 \cdot 0 \sin^{2} (\frac{π}{2}) - 32 \cos^{4} (\frac{π}{2})$

Step 9.1.3

Multiply $96$ by $0$ .

$0 \sin^{2} (\frac{π}{2}) - 32 \cos^{4} (\frac{π}{2})$

Step 9.1.4

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

$0 \cdot 1^{2} - 32 \cos^{4} (\frac{π}{2})$

Step 9.1.5

One to any power is one.

$0 \cdot 1 - 32 \cos^{4} (\frac{π}{2})$

Step 9.1.6

Multiply $0$ by $1$ .

$0 - 32 \cos^{4} (\frac{π}{2})$

Step 9.1.7

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

$0 - 32 \cdot 0^{4}$

Step 9.1.8

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

$0 - 32 \cdot 0$

Step 9.1.9

Multiply $- 32$ by $0$ .

$0 + 0$

Step 9.2

Add $0$ and $0$ .

$0$

Step 10

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

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Step 10.1

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

$(- \infty, 0) \cup (0, \frac{π}{2}) \cup (\frac{π}{2}, π) \cup (π, \frac{3 π}{2}) \cup (\frac{3 π}{2}, \infty)$

Step 10.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, 0)$ in the first derivative $- 32 \cos^{3} (x) \sin (x)$ to check if the result is negative or positive.

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Step 10.2.1

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

$f' (- 2) = - 32 \cos^{3} (- 2) \sin (- 2)$

Step 10.2.2

Simplify the result.

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Step 10.2.2.1

Evaluate $\cos (- 2)$ .

$f' (- 2) = - 32 {(- 0.41614683)}^{3} \sin (- 2)$

Step 10.2.2.2

Raise $- 0.41614683$ to the power of $3$ .

$f' (- 2) = - 32 \cdot (- 0.07206755 \sin (- 2))$

Step 10.2.2.3

Multiply $- 32$ by $- 0.07206755$ .

$f' (- 2) = 2.30616178 \sin (- 2)$

Step 10.2.2.4

Evaluate $\sin (- 2)$ .

$f' (- 2) = 2.30616178 \cdot - 0.90929742$

Step 10.2.2.5

Multiply $2.30616178$ by $- 0.90929742$ .

$f' (- 2) = - 2.09698697$

Step 10.2.2.6

The final answer is $- 2.09698697$ .

$- 2.09698697$

Step 10.3

Substitute any number, such as $1$ , from the interval $(0, \frac{π}{2})$ in the first derivative $- 32 \cos^{3} (x) \sin (x)$ to check if the result is negative or positive.

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Step 10.3.1

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

$f' (1) = - 32 \cos^{3} (1) \sin (1)$

Step 10.3.2

Simplify the result.

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Step 10.3.2.1

Evaluate $\cos (1)$ .

$f' (1) = - 32 \cdot ({0.5403023}^{3} \sin (1))$

Step 10.3.2.2

Raise $0.5403023$ to the power of $3$ .

$f' (1) = - 32 \cdot (0.1577286 \sin (1))$

Step 10.3.2.3

Multiply $- 32$ by $0.1577286$ .

$f' (1) = - 5.04731536 \sin (1)$

Step 10.3.2.4

Evaluate $\sin (1)$ .

$f' (1) = - 5.04731536 \cdot 0.84147098$

Step 10.3.2.5

Multiply $- 5.04731536$ by $0.84147098$ .

$f' (1) = - 4.24716943$

Step 10.3.2.6

The final answer is $- 4.24716943$ .

$- 4.24716943$

Step 10.4

Substitute any number, such as $2$ , from the interval $(\frac{π}{2}, π)$ in the first derivative $- 32 \cos^{3} (x) \sin (x)$ to check if the result is negative or positive.

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Step 10.4.1

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

$f' (2) = - 32 \cos^{3} (2) \sin (2)$

Step 10.4.2

Simplify the result.

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Step 10.4.2.1

Evaluate $\cos (2)$ .

$f' (2) = - 32 {(- 0.41614683)}^{3} \sin (2)$

Step 10.4.2.2

Raise $- 0.41614683$ to the power of $3$ .

$f' (2) = - 32 \cdot (- 0.07206755 \sin (2))$

Step 10.4.2.3

Multiply $- 32$ by $- 0.07206755$ .

$f' (2) = 2.30616178 \sin (2)$

Step 10.4.2.4

Evaluate $\sin (2)$ .

$f' (2) = 2.30616178 \cdot 0.90929742$

Step 10.4.2.5

Multiply $2.30616178$ by $0.90929742$ .

$f' (2) = 2.09698697$

Step 10.4.2.6

The final answer is $2.09698697$ .

$2.09698697$

Step 10.5

Substitute any number, such as $4$ , from the interval $(π, \frac{3 π}{2})$ in the first derivative $- 32 \cos^{3} (x) \sin (x)$ to check if the result is negative or positive.

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Step 10.5.1

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

$f' (4) = - 32 \cos^{3} (4) \sin (4)$

Step 10.5.2

Simplify the result.

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Step 10.5.2.1

Evaluate $\cos (4)$ .

$f' (4) = - 32 {(- 0.65364362)}^{3} \sin (4)$

Step 10.5.2.2

Raise $- 0.65364362$ to the power of $3$ .

$f' (4) = - 32 \cdot (- 0.27926922 \sin (4))$

Step 10.5.2.3

Multiply $- 32$ by $- 0.27926922$ .

$f' (4) = 8.93661523 \sin (4)$

Step 10.5.2.4

Evaluate $\sin (4)$ .

$f' (4) = 8.93661523 \cdot - 0.75680249$

Step 10.5.2.5

Multiply $8.93661523$ by $- 0.75680249$ .

$f' (4) = - 6.7632527$

Step 10.5.2.6

The final answer is $- 6.7632527$ .

$- 6.7632527$

Step 10.6

Substitute any number, such as $7$ , from the interval $(\frac{3 π}{2}, \infty)$ in the first derivative $- 32 \cos^{3} (x) \sin (x)$ to check if the result is negative or positive.

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Step 10.6.1

Replace the variable $x$ with $7$ in the expression.

$f' (7) = - 32 \cos^{3} (7) \sin (7)$

Step 10.6.2

Simplify the result.

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Step 10.6.2.1

Evaluate $\cos (7)$ .

$f' (7) = - 32 \cdot ({0.75390225}^{3} \sin (7))$

Step 10.6.2.2

Raise $0.75390225$ to the power of $3$ .

$f' (7) = - 32 \cdot (0.42849437 \sin (7))$

Step 10.6.2.3

Multiply $- 32$ by $0.42849437$ .

$f' (7) = - 13.71182002 \sin (7)$

Step 10.6.2.4

Evaluate $\sin (7)$ .

$f' (7) = - 13.71182002 \cdot 0.65698659$

Step 10.6.2.5

Multiply $- 13.71182002$ by $0.65698659$ .

$f' (7) = - 9.00848199$

Step 10.6.2.6

The final answer is $- 9.00848199$ .

$- 9.00848199$

Step 10.7

Since the first derivative did not change signs around $x = 0$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 10.8

Since the first derivative changed signs from negative to positive around $x = \frac{π}{2}$ , then $x = \frac{π}{2}$ is a local minimum.

$x = \frac{π}{2}$ is a local minimum

Step 10.9

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

$x = π$ is a local maximum

Step 10.10

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 10.11

These are the local extrema for $f (x) = 8 \cos^{4} (x)$ .

$x = \frac{π}{2}$ is a local minimum

$x = π$ is a local maximum

$x = \frac{π}{2}$ is a local minimum

$x = π$ is a local maximum

Step 11