Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

Evaluate $\frac{d}{d x} [3 \cos (x)]$ .

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $- 1$ by $3$ .

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

Step 1.3

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

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Step 1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos^{3} (x)$ with respect to $x$ is $- \frac{d}{d x} [\cos^{3} (x)]$ .

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

Step 1.3.2

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

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Step 1.3.2.1

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

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

Step 1.3.2.2

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

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

Step 1.3.2.3

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

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

Step 1.3.3

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

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

Step 1.3.4

Multiply $- 1$ by $3$ .

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

Step 1.3.5

Multiply $- 3$ by $- 1$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Reorder terms.

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

Step 1.4.2

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

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Step 1.4.2.1

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

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

Step 1.4.2.2

Factor $3 \sin (x)$ out of $- 3 \sin (x)$ .

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

Step 1.4.2.3

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

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

Step 1.4.3

Reorder $\cos^{2} (x)$ and $- 1$ .

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

Step 1.4.4

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

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

Step 1.4.5

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

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

Step 1.4.6

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

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

Step 1.4.7

Rewrite $- 1 (1 - \cos^{2} (x))$ as $- (1 - \cos^{2} (x))$ .

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

Step 1.4.8

Apply pythagorean identity.

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

Step 1.4.9

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

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Step 1.4.9.1

Move $\sin^{2} (x)$ .

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

Step 1.4.9.2

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

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Step 1.4.9.2.1

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

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

Step 1.4.9.2.2

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

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

Step 1.4.9.3

Add $2$ and $1$ .

$3 \sin^{3} (x) \cdot - 1$

Step 1.4.10

Multiply $- 1$ by $3$ .

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

Step 2

Find the second derivative of the function.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

Multiply $3$ by $- 3$ .

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

Step 2.4

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

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

Step 3

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

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

Step 4

Divide each term in $- 3 \sin^{3} (x) = 0$ by $- 3$ and simplify.

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Step 4.1

Divide each term in $- 3 \sin^{3} (x) = 0$ by $- 3$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 3$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $\sin^{3} (x)$ by $1$ .

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 3$ .

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

$\sin (x) = 0$

Step 7

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

$x = \arcsin (0)$

Step 8

Simplify the right side.

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Step 8.1

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

$x = 0$

Step 9

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 10

Subtract $0$ from $π$ .

$x = π$

Step 11

The solution to the equation $x = 0$ .

$x = 0, π$

Step 12

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 13

Evaluate the second derivative.

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Step 13.1

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

$- 9 \cdot 0^{2} \cos (0)$

Step 13.2

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

$- 9 \cdot 0 \cos (0)$

Step 13.3

Multiply $- 9$ by $0$ .

$0 \cos (0)$

Step 13.4

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

$0 \cdot 1$

Step 13.5

Multiply $0$ by $1$ .

$0$

Step 14

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

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Step 14.1

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

$(- \infty, 0) \cup (0, π) \cup (π, \infty)$

Step 14.2

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

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Step 14.2.1

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

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

Step 14.2.2

Simplify the result.

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Step 14.2.2.1

Evaluate $\sin (- 2)$ .

$f' (- 2) = - 3 {(- 0.90929742)}^{3}$

Step 14.2.2.2

Raise $- 0.90929742$ to the power of $3$ .

$f' (- 2) = - 3 \cdot - 0.75182694$

Step 14.2.2.3

Multiply $- 3$ by $- 0.75182694$ .

$f' (- 2) = 2.25548083$

Step 14.2.2.4

The final answer is $2.25548083$ .

$2.25548083$

Step 14.3

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

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Step 14.3.1

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

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

Step 14.3.2

Simplify the result.

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Step 14.3.2.1

Evaluate $\sin (2)$ .

$f' (2) = - 3 \cdot {0.90929742}^{3}$

Step 14.3.2.2

Raise $0.90929742$ to the power of $3$ .

$f' (2) = - 3 \cdot 0.75182694$

Step 14.3.2.3

Multiply $- 3$ by $0.75182694$ .

$f' (2) = - 2.25548083$

Step 14.3.2.4

The final answer is $- 2.25548083$ .

$- 2.25548083$

Step 14.4

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

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Step 14.4.1

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

$f' (6) = - 3 \sin^{3} (6)$

Step 14.4.2

Simplify the result.

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Step 14.4.2.1

Evaluate $\sin (6)$ .

$f' (6) = - 3 {(- 0.27941549)}^{3}$

Step 14.4.2.2

Raise $- 0.27941549$ to the power of $3$ .

$f' (6) = - 3 \cdot - 0.02181481$

Step 14.4.2.3

Multiply $- 3$ by $- 0.02181481$ .

$f' (6) = 0.06544443$

Step 14.4.2.4

The final answer is $0.06544443$ .

$0.06544443$

Step 14.5

Since the first derivative changed signs from positive to negative around $x = 0$ , then $x = 0$ is a local maximum.

$x = 0$ is a local maximum

Step 14.6

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

$x = π$ is a local minimum

Step 14.7

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

$x = 0$ is a local maximum

$x = π$ is a local minimum

$x = 0$ is a local maximum

$x = π$ is a local minimum

Step 15