Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

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

$\frac{d}{d x} [4 \cos (x)] + \frac{d}{d x} [2 \sin^{2} (x)]$

Step 1.2

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

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Step 1.2.1

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

$4 \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [2 \sin^{2} (x)]$

Step 1.2.2

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

$4 (- \sin (x)) + \frac{d}{d x} [2 \sin^{2} (x)]$

Step 1.2.3

Multiply $- 1$ by $4$ .

$- 4 \sin (x) + \frac{d}{d x} [2 \sin^{2} (x)]$

Step 1.3

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

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Step 1.3.1

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

$- 4 \sin (x) + 2 \frac{d}{d x} [\sin^{2} (x)]$

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

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Step 1.3.2.1

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

$- 4 \sin (x) + 2 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [\sin (x)])$

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 $n = 2$ .

$- 4 \sin (x) + 2 (2 u \frac{d}{d x} [\sin (x)])$

Step 1.3.2.3

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

$- 4 \sin (x) + 2 (2 \sin (x) \frac{d}{d x} [\sin (x)])$

Step 1.3.3

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

$- 4 \sin (x) + 2 (2 \sin (x) \cos (x))$

Step 1.3.4

Multiply $2$ by $2$ .

$- 4 \sin (x) + 4 \sin (x) \cos (x)$

Step 1.4

Reorder terms.

$4 \cos (x) \sin (x) - 4 \sin (x)$

Step 2

Find the second derivative of the function.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

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 $f (x) = \cos (x)$ and $g (x) = \sin (x)$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.2.7

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

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

Step 2.2.8

Add $1$ and $1$ .

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

Step 2.2.9

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

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

Step 2.2.10

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

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

Step 2.2.11

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

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

Step 2.2.12

Add $1$ and $1$ .

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

Step 2.3

Evaluate $\frac{d}{d x} [- 4 \sin (x)]$ .

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Multiply $- 1$ by $4$ .

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

Step 3

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

$4 \cos (x) \sin (x) - 4 \sin (x) = 0$

Step 4

Factor $4 \sin (x)$ out of $4 \cos (x) \sin (x) - 4 \sin (x)$ .

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Step 4.1

Factor $4 \sin (x)$ out of $4 \cos (x) \sin (x)$ .

$4 \sin (x) \cos (x) - 4 \sin (x) = 0$

Step 4.2

Factor $4 \sin (x)$ out of $- 4 \sin (x)$ .

$4 \sin (x) \cos (x) + 4 \sin (x) \cdot - 1 = 0$

Step 4.3

Factor $4 \sin (x)$ out of $4 \sin (x) \cos (x) + 4 \sin (x) \cdot - 1$ .

$4 \sin (x) (\cos (x) - 1) = 0$

Step 5

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

$\sin (x) = 0$

$\cos (x) - 1 = 0$

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

Set $\cos (x) - 1$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $\cos (x) - 1$ equal to $0$ .

$\cos (x) - 1 = 0$

Step 7.2

Solve $\cos (x) - 1 = 0$ for $x$ .

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Step 7.2.1

Add $1$ to both sides of the equation.

$\cos (x) = 1$

Step 7.2.2

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

$x = \arccos (1)$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

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

$x = 0$

Step 7.2.4

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.

$x = 2 π - 0$

Step 7.2.5

Subtract $0$ from $2 π$ .

$x = 2 π$

Step 7.2.6

The solution to the equation $x = 0$ .

$x = 0, 2 π$

Step 8

The final solution is all the values that make $4 \sin (x) (\cos (x) - 1) = 0$ true.

$x = 0, π, 2 π$

Step 9

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.

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

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 $\cos (0)$ is $1$ .

$4 \cdot 1^{2} - 4 \sin^{2} (0) - 4 \cos (0)$

Step 10.1.2

One to any power is one.

$4 \cdot 1 - 4 \sin^{2} (0) - 4 \cos (0)$

Step 10.1.3

Multiply $4$ by $1$ .

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

Step 10.1.4

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

$4 - 4 \cdot 0^{2} - 4 \cos (0)$

Step 10.1.5

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

$4 - 4 \cdot 0 - 4 \cos (0)$

Step 10.1.6

Multiply $- 4$ by $0$ .

$4 + 0 - 4 \cos (0)$

Step 10.1.7

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

$4 + 0 - 4 \cdot 1$

Step 10.1.8

Multiply $- 4$ by $1$ .

$4 + 0 - 4$

Step 10.2

Simplify by adding and subtracting.

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Step 10.2.1

Add $4$ and $0$ .

$4 - 4$

Step 10.2.2

Subtract $4$ from $4$ .

$0$

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.

$(- \infty, 0) \cup (0, π) \cup (π, 2 π) \cup (2 π, \infty)$

Step 11.2

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

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Step 11.2.1

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

$f' (- 2) = 4 \cos (- 2) \sin (- 2) - 4 \sin (- 2)$

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

Evaluate $\cos (- 2)$ .

$f' (- 2) = 4 \cdot (- 0.41614683 \sin (- 2)) - 4 \sin (- 2)$

Step 11.2.2.1.2

Multiply $4$ by $- 0.41614683$ .

$f' (- 2) = - 1.66458734 \sin (- 2) - 4 \sin (- 2)$

Step 11.2.2.1.3

Evaluate $\sin (- 2)$ .

$f' (- 2) = - 1.66458734 \cdot - 0.90929742 - 4 \sin (- 2)$

Step 11.2.2.1.4

Multiply $- 1.66458734$ by $- 0.90929742$ .

$f' (- 2) = 1.51360499 - 4 \sin (- 2)$

Step 11.2.2.1.5

Evaluate $\sin (- 2)$ .

$f' (- 2) = 1.51360499 - 4 \cdot - 0.90929742$

Step 11.2.2.1.6

Multiply $- 4$ by $- 0.90929742$ .

$f' (- 2) = 1.51360499 + 3.6371897$

Step 11.2.2.2

Add $1.51360499$ and $3.6371897$ .

$f' (- 2) = 5.15079469$

Step 11.2.2.3

The final answer is $5.15079469$ .

$5.15079469$

Step 11.3

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

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Step 11.3.1

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

$f' (2) = 4 \cos (2) \sin (2) - 4 \sin (2)$

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 $\cos (2)$ .

$f' (2) = 4 \cdot (- 0.41614683 \sin (2)) - 4 \sin (2)$

Step 11.3.2.1.2

Multiply $4$ by $- 0.41614683$ .

$f' (2) = - 1.66458734 \sin (2) - 4 \sin (2)$

Step 11.3.2.1.3

Evaluate $\sin (2)$ .

$f' (2) = - 1.66458734 \cdot 0.90929742 - 4 \sin (2)$

Step 11.3.2.1.4

Multiply $- 1.66458734$ by $0.90929742$ .

$f' (2) = - 1.51360499 - 4 \sin (2)$

Step 11.3.2.1.5

Evaluate $\sin (2)$ .

$f' (2) = - 1.51360499 - 4 \cdot 0.90929742$

Step 11.3.2.1.6

Multiply $- 4$ by $0.90929742$ .

$f' (2) = - 1.51360499 - 3.6371897$

Step 11.3.2.2

Subtract $3.6371897$ from $- 1.51360499$ .

$f' (2) = - 5.15079469$

Step 11.3.2.3

The final answer is $- 5.15079469$ .

$- 5.15079469$

Step 11.4

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

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Step 11.4.1

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

$f' (5) = 4 \cos (5) \sin (5) - 4 \sin (5)$

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 $\cos (5)$ .

$f' (5) = 4 \cdot (0.28366218 \sin (5)) - 4 \sin (5)$

Step 11.4.2.1.2

Multiply $4$ by $0.28366218$ .

$f' (5) = 1.13464874 \sin (5) - 4 \sin (5)$

Step 11.4.2.1.3

Evaluate $\sin (5)$ .

$f' (5) = 1.13464874 \cdot - 0.95892427 - 4 \sin (5)$

Step 11.4.2.1.4

Multiply $1.13464874$ by $- 0.95892427$ .

$f' (5) = - 1.08804222 - 4 \sin (5)$

Step 11.4.2.1.5

Evaluate $\sin (5)$ .

$f' (5) = - 1.08804222 - 4 \cdot - 0.95892427$

Step 11.4.2.1.6

Multiply $- 4$ by $- 0.95892427$ .

$f' (5) = - 1.08804222 + 3.83569709$

Step 11.4.2.2

Add $- 1.08804222$ and $3.83569709$ .

$f' (5) = 2.74765487$

Step 11.4.2.3

The final answer is $2.74765487$ .

$2.74765487$

Step 11.5

Substitute any number, such as $9$ , from the interval $(2 π, \infty)$ in the first derivative $4 \cos (x) \sin (x) - 4 \sin (x)$ to check if the result is negative or positive.

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Step 11.5.1

Replace the variable $x$ with $9$ in the expression.

$f' (9) = 4 \cos (9) \sin (9) - 4 \sin (9)$

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 $\cos (9)$ .

$f' (9) = 4 \cdot (- 0.91113026 \sin (9)) - 4 \sin (9)$

Step 11.5.2.1.2

Multiply $4$ by $- 0.91113026$ .

$f' (9) = - 3.64452104 \sin (9) - 4 \sin (9)$

Step 11.5.2.1.3

Evaluate $\sin (9)$ .

$f' (9) = - 3.64452104 \cdot 0.41211848 - 4 \sin (9)$

Step 11.5.2.1.4

Multiply $- 3.64452104$ by $0.41211848$ .

$f' (9) = - 1.50197449 - 4 \sin (9)$

Step 11.5.2.1.5

Evaluate $\sin (9)$ .

$f' (9) = - 1.50197449 - 4 \cdot 0.41211848$

Step 11.5.2.1.6

Multiply $- 4$ by $0.41211848$ .

$f' (9) = - 1.50197449 - 1.64847394$

Step 11.5.2.2

Subtract $1.64847394$ from $- 1.50197449$ .

$f' (9) = - 3.15044843$

Step 11.5.2.3

The final answer is $- 3.15044843$ .

$- 3.15044843$

Step 11.6

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 11.7

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 11.8

Since the first derivative changed signs from positive to negative around $x = 2 π$ , then $x = 2 π$ is a local maximum.

$x = 2 π$ is a local maximum

Step 11.9

These are the local extrema for $f (x) = 4 \cos (x) + 2 \sin^{2} (x)$ .

$x = 0$ is a local maximum

$x = π$ is a local minimum

$x = 2 π$ is a local maximum

$x = 0$ is a local maximum

$x = π$ is a local minimum

$x = 2 π$ is a local maximum

Step 12