Calculus Examples

$f (x) = 2 \cos (x) + \sin (2 x)$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of

$2 \cos (x) + \sin (2 x)$ with respect to

$x$ is

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

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

Step 1.2

Evaluate

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

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Step 1.2.1

Since

$2$ is constant with respect to

$x$ , the derivative of

$2 \cos (x)$ with respect to

$x$ is

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

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

Step 1.2.2

The derivative of

$\cos (x)$ with respect to

$x$ is

$- \sin (x)$ .

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

Step 1.2.3

Multiply

$- 1$ by

$2$ .

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

Step 1.3

Evaluate

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

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Step 1.3.1

Differentiate using the chain rule, which states that

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = \sin (x)$ and

$g (x) = 2 x$ .

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Step 1.3.1.1

To apply the Chain Rule, set

$u$ as

$2 x$ .

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

Step 1.3.1.2

The derivative of

$\sin (u)$ with respect to

$u$ is

$\cos (u)$ .

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

Step 1.3.1.3

Replace all occurrences of

$u$ with

$2 x$ .

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

Step 1.3.2

Since

$2$ is constant with respect to

$x$ , the derivative of

$2 x$ with respect to

$x$ is

$2 \frac{d}{d x} [x]$ .

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

Step 1.3.3

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$- 2 \sin (x) + \cos (2 x) (2 \cdot 1)$

Step 1.3.4

Multiply

$2$ by

$1$ .

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

Step 1.3.5

Move

$2$ to the left of

$\cos (2 x)$ .

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

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of

$- 2 \sin (x) + 2 \cos (2 x)$ with respect to

$x$ is

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

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

Step 2.2

Evaluate

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

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Step 2.2.1

Since

$- 2$ is constant with respect to

$x$ , the derivative of

$- 2 \sin (x)$ with respect to

$x$ is

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

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

Step 2.2.2

The derivative of

$\sin (x)$ with respect to

$x$ is

$\cos (x)$ .

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

Step 2.3

Evaluate

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

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Step 2.3.1

Since

$2$ is constant with respect to

$x$ , the derivative of

$2 \cos (2 x)$ with respect to

$x$ is

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

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

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

$f (x) = \cos (x)$ and

$g (x) = 2 x$ .

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Step 2.3.2.1

To apply the Chain Rule, set

$u$ as

$2 x$ .

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

Step 2.3.2.2

The derivative of

$\cos (u)$ with respect to

$u$ is

$- \sin (u)$ .

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

Step 2.3.2.3

Replace all occurrences of

$u$ with

$2 x$ .

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

Step 2.3.3

Since

$2$ is constant with respect to

$x$ , the derivative of

$2 x$ with respect to

$x$ is

$2 \frac{d}{d x} [x]$ .

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

Step 2.3.4

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$f'' (x) = - 2 \cos (x) + 2 (- \sin (2 x) (2 \cdot 1))$

Step 2.3.5

Multiply

$2$ by

$1$ .

$f'' (x) = - 2 \cos (x) + 2 (- \sin (2 x) \cdot 2)$

Step 2.3.6

Multiply

$2$ by

$- 1$ .

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

Step 2.3.7

Multiply

$- 2$ by

$2$ .

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

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to

$0$ and solve.

$- 2 \sin (x) + 2 \cos (2 x) = 0$

Step 4

Simplify each term.

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Step 4.1

Use the double-angle identity to transform

$\cos (2 x)$ to

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

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Multiply

$2$ by

$1$ .

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

Step 4.4

Multiply

$- 2$ by

$2$ .

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

Step 5

Factor

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

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Step 5.1

Factor

$2$ out of

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

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Step 5.1.1

Factor

$2$ out of

$- 2 \sin (x)$ .

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

Step 5.1.2

Factor

$2$ out of

$2$ .

$2 (- \sin (x)) + 2 (1) - 4 \sin^{2} (x) = 0$

Step 5.1.3

Factor

$2$ out of

$- 4 \sin^{2} (x)$ .

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

Step 5.1.4

Factor

$2$ out of

$2 (- \sin (x)) + 2 (1)$ .

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

Step 5.1.5

Factor

$2$ out of

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

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

Step 5.2

Factor.

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Step 5.2.1

Factor by grouping.

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Step 5.2.1.1

Reorder terms.

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

Step 5.2.1.2

For a polynomial of the form

$a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is

$a \cdot c = - 2 \cdot 1 = - 2$ and whose sum is

$b = - 1$ .

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Step 5.2.1.2.1

Factor

$- 1$ out of

$- \sin (x)$ .

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

Step 5.2.1.2.2

Rewrite

$- 1$ as

$1$ plus

$- 2$

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

Step 5.2.1.2.3

Apply the distributive property.

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

Step 5.2.1.2.4

Multiply

$\sin (x)$ by

$1$ .

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

Step 5.2.1.3

Factor out the greatest common factor from each group.

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Step 5.2.1.3.1

Group the first two terms and the last two terms.

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

Step 5.2.1.3.2

Factor out the greatest common factor (GCF) from each group.

$2 (\sin (x) (- 2 \sin (x) + 1) + 1 (- 2 \sin (x) + 1)) = 0$

Step 5.2.1.4

Factor the polynomial by factoring out the greatest common factor,

$- 2 \sin (x) + 1$ .

$2 ((- 2 \sin (x) + 1) (\sin (x) + 1)) = 0$

Step 5.2.2

Remove unnecessary parentheses.

$2 (- 2 \sin (x) + 1) (\sin (x) + 1) = 0$

Step 6

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$- 2 \sin (x) + 1 = 0$

$\sin (x) + 1 = 0$

Step 7

Set

$- 2 \sin (x) + 1$ equal to

$0$ and solve for

$x$ .

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Step 7.1

Set

$- 2 \sin (x) + 1$ equal to

$0$ .

$- 2 \sin (x) + 1 = 0$

Step 7.2

Solve

$- 2 \sin (x) + 1 = 0$ for

$x$ .

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Step 7.2.1

Subtract

$1$ from both sides of the equation.

$- 2 \sin (x) = - 1$

Step 7.2.2

Divide each term in

$- 2 \sin (x) = - 1$ by

$- 2$ and simplify.

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Step 7.2.2.1

Divide each term in

$- 2 \sin (x) = - 1$ by

$- 2$ .

$\frac{- 2 \sin (x)}{- 2} = \frac{- 1}{- 2}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of

$- 2$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{- 2 \sin (x)}{- 2} = \frac{- 1}{- 2}$

Step 7.2.2.2.1.2

Divide

$\sin (x)$ by

$1$ .

$\sin (x) = \frac{- 1}{- 2}$

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Dividing two negative values results in a positive value.

$\sin (x) = \frac{1}{2}$

Step 7.2.3

Take the inverse sine of both sides of the equation to extract

$x$ from inside the sine.

$x = \arcsin (\frac{1}{2})$

Step 7.2.4

Simplify the right side.

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Step 7.2.4.1

The exact value of

$\arcsin (\frac{1}{2})$ is

$\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 7.2.5

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.

$x = π - \frac{π}{6}$

Step 7.2.6

Simplify

$π - \frac{π}{6}$ .

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Step 7.2.6.1

To write

$π$ as a fraction with a common denominator, multiply by

$\frac{6}{6}$ .

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

Step 7.2.6.2

Combine fractions.

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Step 7.2.6.2.1

Combine

$π$ and

$\frac{6}{6}$ .

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

Step 7.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 - π}{6}$

Step 7.2.6.3

Simplify the numerator.

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Step 7.2.6.3.1

Move

$6$ to the left of

$π$ .

$x = \frac{6 \cdot π - π}{6}$

Step 7.2.6.3.2

Subtract

$π$ from

$6 π$ .

$x = \frac{5 π}{6}$

Step 7.2.7

The solution to the equation

$x = \frac{π}{6}$ .

$x = \frac{π}{6}, \frac{5 π}{6}$

Step 8

Set

$\sin (x) + 1$ equal to

$0$ and solve for

$x$ .

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Step 8.1

Set

$\sin (x) + 1$ equal to

$0$ .

$\sin (x) + 1 = 0$

Step 8.2

Solve

$\sin (x) + 1 = 0$ for

$x$ .

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Step 8.2.1

Subtract

$1$ from both sides of the equation.

$\sin (x) = - 1$

Step 8.2.2

Take the inverse sine of both sides of the equation to extract

$x$ from inside the sine.

$x = \arcsin (- 1)$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

The exact value of

$\arcsin (- 1)$ is

$- \frac{π}{2}$ .

$x = - \frac{π}{2}$

Step 8.2.4

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from

$2 π$ , to find a reference angle. Next, add this reference angle to

$π$ to find the solution in the third quadrant.

$x = 2 π + \frac{π}{2} + π$

Step 8.2.5

Simplify the expression to find the second solution.

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Step 8.2.5.1

Subtract

$2 π$ from

$2 π + \frac{π}{2} + π$ .

$x = 2 π + \frac{π}{2} + π - 2 π$

Step 8.2.5.2

The resulting angle of

$\frac{3 π}{2}$ is positive, less than

$2 π$ , and coterminal with

$2 π + \frac{π}{2} + π$ .

$x = \frac{3 π}{2}$

Step 8.2.6

The solution to the equation

$x = - \frac{π}{2}$ .

$x = - \frac{π}{2}, \frac{3 π}{2}$

Step 9

The final solution is all the values that make

$2 (- 2 \sin (x) + 1) (\sin (x) + 1) = 0$ true.

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

Step 10

Evaluate the second derivative at

$x = \frac{π}{6}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 2 \cos (\frac{π}{6}) - 4 \sin (2 (\frac{π}{6}))$

Step 11

Evaluate the second derivative.

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Step 11.1

Simplify each term.

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Step 11.1.1

The exact value of

$\cos (\frac{π}{6})$ is

$\frac{\sqrt{3}}{2}$ .

$- 2 \frac{\sqrt{3}}{2} - 4 \sin (2 (\frac{π}{6}))$

Step 11.1.2

Cancel the common factor of

$2$ .

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Step 11.1.2.1

Factor

$2$ out of

$- 2$ .

$2 (- 1) \frac{\sqrt{3}}{2} - 4 \sin (2 (\frac{π}{6}))$

Step 11.1.2.2

Cancel the common factor.

$2 \cdot - 1 \frac{\sqrt{3}}{2} - 4 \sin (2 (\frac{π}{6}))$

Step 11.1.2.3

Rewrite the expression.

$- 1 \sqrt{3} - 4 \sin (2 (\frac{π}{6}))$

Step 11.1.3

Rewrite

$- 1 \sqrt{3}$ as

$- \sqrt{3}$ .

$- \sqrt{3} - 4 \sin (2 (\frac{π}{6}))$

Step 11.1.4

Cancel the common factor of

$2$ .

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Step 11.1.4.1

Factor

$2$ out of

$6$ .

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

Step 11.1.4.2

Cancel the common factor.

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

Step 11.1.4.3

Rewrite the expression.

$- \sqrt{3} - 4 \sin (\frac{π}{3})$

Step 11.1.5

The exact value of

$\sin (\frac{π}{3})$ is

$\frac{\sqrt{3}}{2}$ .

$- \sqrt{3} - 4 \frac{\sqrt{3}}{2}$

Step 11.1.6

Cancel the common factor of

$2$ .

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Step 11.1.6.1

Factor

$2$ out of

$- 4$ .

$- \sqrt{3} + 2 (- 2) \frac{\sqrt{3}}{2}$

Step 11.1.6.2

Cancel the common factor.

$- \sqrt{3} + 2 \cdot - 2 \frac{\sqrt{3}}{2}$

Step 11.1.6.3

Rewrite the expression.

$- \sqrt{3} - 2 \sqrt{3}$

Step 11.2

Subtract

$2 \sqrt{3}$ from

$- \sqrt{3}$ .

$- 3 \sqrt{3}$

Step 12

$x = \frac{π}{6}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{π}{6}$ is a local maximum

Step 13

Find the y-value when

$x = \frac{π}{6}$ .

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Step 13.1

Replace the variable

$x$ with

$\frac{π}{6}$ in the expression.

$f (\frac{π}{6}) = 2 \cos (\frac{π}{6}) + \sin (2 (\frac{π}{6}))$

Step 13.2

Simplify the result.

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Step 13.2.1

Simplify each term.

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Step 13.2.1.1

The exact value of

$\cos (\frac{π}{6})$ is

$\frac{\sqrt{3}}{2}$ .

$f (\frac{π}{6}) = 2 (\frac{\sqrt{3}}{2}) + \sin (2 (\frac{π}{6}))$

Step 13.2.1.2

Cancel the common factor of

$2$ .

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Step 13.2.1.2.1

Cancel the common factor.

$f (\frac{π}{6}) = 2 (\frac{\sqrt{3}}{2}) + \sin (2 (\frac{π}{6}))$

Step 13.2.1.2.2

Rewrite the expression.

$f (\frac{π}{6}) = \sqrt{3} + \sin (2 (\frac{π}{6}))$

Step 13.2.1.3

Cancel the common factor of

$2$ .

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Step 13.2.1.3.1

Factor

$2$ out of

$6$ .

$f (\frac{π}{6}) = \sqrt{3} + \sin (2 (\frac{π}{2 (3)}))$

Step 13.2.1.3.2

Cancel the common factor.

$f (\frac{π}{6}) = \sqrt{3} + \sin (2 (\frac{π}{2 \cdot 3}))$

Step 13.2.1.3.3

Rewrite the expression.

$f (\frac{π}{6}) = \sqrt{3} + \sin (\frac{π}{3})$

Step 13.2.1.4

The exact value of

$\sin (\frac{π}{3})$ is

$\frac{\sqrt{3}}{2}$ .

$f (\frac{π}{6}) = \sqrt{3} + \frac{\sqrt{3}}{2}$

Step 13.2.2

To write

$\sqrt{3}$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$f (\frac{π}{6}) = \sqrt{3} \cdot \frac{2}{2} + \frac{\sqrt{3}}{2}$

Step 13.2.3

Combine fractions.

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Step 13.2.3.1

Combine

$\sqrt{3}$ and

$\frac{2}{2}$ .

$f (\frac{π}{6}) = \frac{\sqrt{3} \cdot 2}{2} + \frac{\sqrt{3}}{2}$

Step 13.2.3.2

Combine the numerators over the common denominator.

$f (\frac{π}{6}) = \frac{\sqrt{3} \cdot 2 + \sqrt{3}}{2}$

Step 13.2.4

Simplify the numerator.

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Step 13.2.4.1

Move

$2$ to the left of

$\sqrt{3}$ .

$f (\frac{π}{6}) = \frac{2 \cdot \sqrt{3} + \sqrt{3}}{2}$

Step 13.2.4.2

Add

$2 \sqrt{3}$ and

$\sqrt{3}$ .

$f (\frac{π}{6}) = \frac{3 \sqrt{3}}{2}$

Step 13.2.5

The final answer is

$\frac{3 \sqrt{3}}{2}$ .

$y = \frac{3 \sqrt{3}}{2}$

Step 14

Evaluate the second derivative at

$x = \frac{5 π}{6}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 2 \cos (\frac{5 π}{6}) - 4 \sin (2 (\frac{5 π}{6}))$

Step 15

Evaluate the second derivative.

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Step 15.1

Simplify each term.

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Step 15.1.1

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.

$- 2 (- \cos (\frac{π}{6})) - 4 \sin (2 (\frac{5 π}{6}))$

Step 15.1.2

The exact value of

$\cos (\frac{π}{6})$ is

$\frac{\sqrt{3}}{2}$ .

$- 2 (- \frac{\sqrt{3}}{2}) - 4 \sin (2 (\frac{5 π}{6}))$

Step 15.1.3

Cancel the common factor of

$2$ .

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Step 15.1.3.1

Move the leading negative in

$- \frac{\sqrt{3}}{2}$ into the numerator.

$- 2 \frac{- \sqrt{3}}{2} - 4 \sin (2 (\frac{5 π}{6}))$

Step 15.1.3.2

Factor

$2$ out of

$- 2$ .

$2 (- 1) \frac{- \sqrt{3}}{2} - 4 \sin (2 (\frac{5 π}{6}))$

Step 15.1.3.3

Cancel the common factor.

$2 \cdot - 1 \frac{- \sqrt{3}}{2} - 4 \sin (2 (\frac{5 π}{6}))$

Step 15.1.3.4

Rewrite the expression.

$- 1 (- \sqrt{3}) - 4 \sin (2 (\frac{5 π}{6}))$

Step 15.1.4

Multiply

$- 1$ by

$- 1$ .

$1 \sqrt{3} - 4 \sin (2 (\frac{5 π}{6}))$

Step 15.1.5

Multiply

$\sqrt{3}$ by

$1$ .

$\sqrt{3} - 4 \sin (2 (\frac{5 π}{6}))$

Step 15.1.6

Cancel the common factor of

$2$ .

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Step 15.1.6.1

Factor

$2$ out of

$6$ .

$\sqrt{3} - 4 \sin (2 \frac{5 π}{2 (3)})$

Step 15.1.6.2

Cancel the common factor.

$\sqrt{3} - 4 \sin (2 \frac{5 π}{2 \cdot 3})$

Step 15.1.6.3

Rewrite the expression.

$\sqrt{3} - 4 \sin (\frac{5 π}{3})$

Step 15.1.7

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.

$\sqrt{3} - 4 (- \sin (\frac{π}{3}))$

Step 15.1.8

The exact value of

$\sin (\frac{π}{3})$ is

$\frac{\sqrt{3}}{2}$ .

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

Step 15.1.9

Cancel the common factor of

$2$ .

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Step 15.1.9.1

Move the leading negative in

$- \frac{\sqrt{3}}{2}$ into the numerator.

$\sqrt{3} - 4 \frac{- \sqrt{3}}{2}$

Step 15.1.9.2

Factor

$2$ out of

$- 4$ .

$\sqrt{3} + 2 (- 2) \frac{- \sqrt{3}}{2}$

Step 15.1.9.3

Cancel the common factor.

$\sqrt{3} + 2 \cdot - 2 \frac{- \sqrt{3}}{2}$

Step 15.1.9.4

Rewrite the expression.

$\sqrt{3} - 2 (- \sqrt{3})$

Step 15.1.10

Multiply

$- 1$ by

$- 2$ .

$\sqrt{3} + 2 \sqrt{3}$

Step 15.2

Add

$\sqrt{3}$ and

$2 \sqrt{3}$ .

$3 \sqrt{3}$

Step 16

$x = \frac{5 π}{6}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

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

Step 17

Find the y-value when

$x = \frac{5 π}{6}$ .

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Step 17.1

Replace the variable

$x$ with

$\frac{5 π}{6}$ in the expression.

$f (\frac{5 π}{6}) = 2 \cos (\frac{5 π}{6}) + \sin (2 (\frac{5 π}{6}))$

Step 17.2

Simplify the result.

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Step 17.2.1

Simplify each term.

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Step 17.2.1.1

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.

$f (\frac{5 π}{6}) = 2 (- \cos (\frac{π}{6})) + \sin (2 (\frac{5 π}{6}))$

Step 17.2.1.2

The exact value of

$\cos (\frac{π}{6})$ is

$\frac{\sqrt{3}}{2}$ .

$f (\frac{5 π}{6}) = 2 (- \frac{\sqrt{3}}{2}) + \sin (2 (\frac{5 π}{6}))$

Step 17.2.1.3

Cancel the common factor of

$2$ .

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Step 17.2.1.3.1

Move the leading negative in

$- \frac{\sqrt{3}}{2}$ into the numerator.

$f (\frac{5 π}{6}) = 2 (\frac{- \sqrt{3}}{2}) + \sin (2 (\frac{5 π}{6}))$

Step 17.2.1.3.2

Cancel the common factor.

$f (\frac{5 π}{6}) = 2 (\frac{- \sqrt{3}}{2}) + \sin (2 (\frac{5 π}{6}))$

Step 17.2.1.3.3

Rewrite the expression.

$f (\frac{5 π}{6}) = - \sqrt{3} + \sin (2 (\frac{5 π}{6}))$

Step 17.2.1.4

Cancel the common factor of

$2$ .

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Step 17.2.1.4.1

Factor

$2$ out of

$6$ .

$f (\frac{5 π}{6}) = - \sqrt{3} + \sin (2 (\frac{5 π}{2 (3)}))$

Step 17.2.1.4.2

Cancel the common factor.

$f (\frac{5 π}{6}) = - \sqrt{3} + \sin (2 (\frac{5 π}{2 \cdot 3}))$

Step 17.2.1.4.3

Rewrite the expression.

$f (\frac{5 π}{6}) = - \sqrt{3} + \sin (\frac{5 π}{3})$

Step 17.2.1.5

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 fourth quadrant.

$f (\frac{5 π}{6}) = - \sqrt{3} - \sin (\frac{π}{3})$

Step 17.2.1.6

The exact value of

$\sin (\frac{π}{3})$ is

$\frac{\sqrt{3}}{2}$ .

$f (\frac{5 π}{6}) = - \sqrt{3} - \frac{\sqrt{3}}{2}$

Step 17.2.2

To write

$- \sqrt{3}$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$f (\frac{5 π}{6}) = - \sqrt{3} \cdot \frac{2}{2} - \frac{\sqrt{3}}{2}$

Step 17.2.3

Combine fractions.

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Step 17.2.3.1

Combine

$- \sqrt{3}$ and

$\frac{2}{2}$ .

$f (\frac{5 π}{6}) = \frac{- \sqrt{3} \cdot 2}{2} - \frac{\sqrt{3}}{2}$

Step 17.2.3.2

Combine the numerators over the common denominator.

$f (\frac{5 π}{6}) = \frac{- \sqrt{3} \cdot 2 - \sqrt{3}}{2}$

Step 17.2.4

Simplify the numerator.

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Step 17.2.4.1

Multiply

$2$ by

$- 1$ .

$f (\frac{5 π}{6}) = \frac{- 2 \sqrt{3} - \sqrt{3}}{2}$

Step 17.2.4.2

Subtract

$\sqrt{3}$ from

$- 2 \sqrt{3}$ .

$f (\frac{5 π}{6}) = \frac{- 3 \sqrt{3}}{2}$

Step 17.2.5

Move the negative in front of the fraction.

$f (\frac{5 π}{6}) = - \frac{3 \sqrt{3}}{2}$

Step 17.2.6

The final answer is

$- \frac{3 \sqrt{3}}{2}$ .

$y = - \frac{3 \sqrt{3}}{2}$

Step 18

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.

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

Step 19

Evaluate the second derivative.

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Step 19.1

Simplify each term.

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Step 19.1.1

Add full rotations of

$2 π$ until the angle is greater than or equal to

$0$ and less than

$2 π$ .

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

Step 19.1.2

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

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

Step 19.1.3

The exact value of

$\cos (\frac{π}{2})$ is

$0$ .

$- 2 \cdot 0 - 4 \sin (2 (- \frac{π}{2}))$

Step 19.1.4

Multiply

$- 2$ by

$0$ .

$0 - 4 \sin (2 (- \frac{π}{2}))$

Step 19.1.5

Cancel the common factor of

$2$ .

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Step 19.1.5.1

Move the leading negative in

$- \frac{π}{2}$ into the numerator.

$0 - 4 \sin (2 \frac{- π}{2})$

Step 19.1.5.2

Cancel the common factor.

$0 - 4 \sin (2 \frac{- π}{2})$

Step 19.1.5.3

Rewrite the expression.

$0 - 4 \sin (- π)$

Step 19.1.6

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

$0 - 4 \sin (0)$

Step 19.1.7

The exact value of

$\sin (0)$ is

$0$ .

$0 - 4 \cdot 0$

Step 19.1.8

Multiply

$- 4$ by

$0$ .

$0 + 0$

Step 19.2

Add

$0$ and

$0$ .

$0$

Step 20

Since there is at least one point with

$0$ or undefined second derivative, apply the first derivative test.

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Step 20.1

Split

$(- \infty, \infty)$ into separate intervals around the

$x$ values that make the first derivative

$0$ or undefined.

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

Step 20.2

Substitute any number, such as

$- 4$ , from the interval

$(- \infty, - \frac{π}{2})$ in the first derivative

$- 2 \sin (x) + 2 \cos (2 x)$ to check if the result is negative or positive.

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Step 20.2.1

Replace the variable

$x$ with

$- 4$ in the expression.

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

Step 20.2.2

Simplify the result.

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Step 20.2.2.1

Simplify each term.

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Step 20.2.2.1.1

Evaluate

$\sin (- 4)$ .

$f' (- 4) = - 2 \cdot 0.75680249 + 2 \cos (2 (- 4))$

Step 20.2.2.1.2

Multiply

$- 2$ by

$0.75680249$ .

$f' (- 4) = - 1.51360499 + 2 \cos (2 (- 4))$

Step 20.2.2.1.3

Multiply

$2$ by

$- 4$ .

$f' (- 4) = - 1.51360499 + 2 \cos (- 8)$

Step 20.2.2.1.4

Evaluate

$\cos (- 8)$ .

$f' (- 4) = - 1.51360499 + 2 \cdot - 0.14550003$

Step 20.2.2.1.5

Multiply

$2$ by

$- 0.14550003$ .

$f' (- 4) = - 1.51360499 - 0.29100006$

Step 20.2.2.2

Subtract

$0.29100006$ from

$- 1.51360499$ .

$f' (- 4) = - 1.80460505$

Step 20.2.2.3

The final answer is

$- 1.80460505$ .

$- 1.80460505$

Step 20.3

Substitute any number, such as

$0$ , from the interval

$(- \frac{π}{2}, \frac{π}{6})$ in the first derivative

$- 2 \sin (x) + 2 \cos (2 x)$ to check if the result is negative or positive.

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Step 20.3.1

Replace the variable

$x$ with

$0$ in the expression.

$f' (0) = - 2 \sin (0) + 2 \cos (2 (0))$

Step 20.3.2

Simplify the result.

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Step 20.3.2.1

Simplify each term.

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Step 20.3.2.1.1

The exact value of

$\sin (0)$ is

$0$ .

$f' (0) = - 2 \cdot 0 + 2 \cos (2 (0))$

Step 20.3.2.1.2

Multiply

$- 2$ by

$0$ .

$f' (0) = 0 + 2 \cos (2 (0))$

Step 20.3.2.1.3

Multiply

$2$ by

$0$ .

$f' (0) = 0 + 2 \cos (0)$

Step 20.3.2.1.4

The exact value of

$\cos (0)$ is

$1$ .

$f' (0) = 0 + 2 \cdot 1$

Step 20.3.2.1.5

Multiply

$2$ by

$1$ .

$f' (0) = 0 + 2$

Step 20.3.2.2

Add

$0$ and

$2$ .

$f' (0) = 2$

Step 20.3.2.3

The final answer is

$2$ .

$2$

Step 20.4

Substitute any number, such as

$2$ , from the interval

$(\frac{π}{6}, \frac{5 π}{6})$ in the first derivative

$- 2 \sin (x) + 2 \cos (2 x)$ to check if the result is negative or positive.

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Step 20.4.1

Replace the variable

$x$ with

$2$ in the expression.

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

Step 20.4.2

Simplify the result.

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Step 20.4.2.1

Simplify each term.

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Step 20.4.2.1.1

Evaluate

$\sin (2)$ .

$f' (2) = - 2 \cdot 0.90929742 + 2 \cos (2 (2))$

Step 20.4.2.1.2

Multiply

$- 2$ by

$0.90929742$ .

$f' (2) = - 1.81859485 + 2 \cos (2 (2))$

Step 20.4.2.1.3

Multiply

$2$ by

$2$ .

$f' (2) = - 1.81859485 + 2 \cos (4)$

Step 20.4.2.1.4

Evaluate

$\cos (4)$ .

$f' (2) = - 1.81859485 + 2 \cdot - 0.65364362$

Step 20.4.2.1.5

Multiply

$2$ by

$- 0.65364362$ .

$f' (2) = - 1.81859485 - 1.30728724$

Step 20.4.2.2

Subtract

$1.30728724$ from

$- 1.81859485$ .

$f' (2) = - 3.12588209$

Step 20.4.2.3

The final answer is

$- 3.12588209$ .

$- 3.12588209$

Step 20.5

Substitute any number, such as

$4$ , from the interval

$(\frac{5 π}{6}, \frac{3 π}{2})$ in the first derivative

$- 2 \sin (x) + 2 \cos (2 x)$ to check if the result is negative or positive.

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Step 20.5.1

Replace the variable

$x$ with

$4$ in the expression.

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

Step 20.5.2

Simplify the result.

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Step 20.5.2.1

Simplify each term.

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Step 20.5.2.1.1

Evaluate

$\sin (4)$ .

$f' (4) = - 2 \cdot - 0.75680249 + 2 \cos (2 (4))$

Step 20.5.2.1.2

Multiply

$- 2$ by

$- 0.75680249$ .

$f' (4) = 1.51360499 + 2 \cos (2 (4))$

Step 20.5.2.1.3

Multiply

$2$ by

$4$ .

$f' (4) = 1.51360499 + 2 \cos (8)$

Step 20.5.2.1.4

Evaluate

$\cos (8)$ .

$f' (4) = 1.51360499 + 2 \cdot - 0.14550003$

Step 20.5.2.1.5

Multiply

$2$ by

$- 0.14550003$ .

$f' (4) = 1.51360499 - 0.29100006$

Step 20.5.2.2

Subtract

$0.29100006$ from

$1.51360499$ .

$f' (4) = 1.22260492$

Step 20.5.2.3

The final answer is

$1.22260492$ .

$1.22260492$

Step 20.6

Substitute any number, such as

$7$ , from the interval

$(\frac{3 π}{2}, \infty)$ in the first derivative

$- 2 \sin (x) + 2 \cos (2 x)$ to check if the result is negative or positive.

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Step 20.6.1

Replace the variable

$x$ with

$7$ in the expression.

$f' (7) = - 2 \sin (7) + 2 \cos (2 (7))$

Step 20.6.2

Simplify the result.

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Step 20.6.2.1

Simplify each term.

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Step 20.6.2.1.1

Evaluate

$\sin (7)$ .

$f' (7) = - 2 \cdot 0.65698659 + 2 \cos (2 (7))$

Step 20.6.2.1.2

Multiply

$- 2$ by

$0.65698659$ .

$f' (7) = - 1.31397319 + 2 \cos (2 (7))$

Step 20.6.2.1.3

Multiply

$2$ by

$7$ .

$f' (7) = - 1.31397319 + 2 \cos (14)$

Step 20.6.2.1.4

Evaluate

$\cos (14)$ .

$f' (7) = - 1.31397319 + 2 \cdot 0.13673721$

Step 20.6.2.1.5

Multiply

$2$ by

$0.13673721$ .

$f' (7) = - 1.31397319 + 0.27347443$

Step 20.6.2.2

Add

$- 1.31397319$ and

$0.27347443$ .

$f' (7) = - 1.04049876$

Step 20.6.2.3

The final answer is

$- 1.04049876$ .

$- 1.04049876$

Step 20.7

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 20.8

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 20.9

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 20.10

Since the first derivative changed signs from positive to negative around

$x = \frac{3 π}{2}$ , then

$x = \frac{3 π}{2}$ is a local maximum.

$x = \frac{3 π}{2}$ is a local maximum

Step 20.11

These are the local extrema for

$f (x) = 2 \cos (x) + \sin (2 x)$ .

$x = - \frac{π}{2}$ is a local minimum

$x = \frac{π}{6}$ is a local maximum

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

$x = \frac{3 π}{2}$ is a local maximum

$x = - \frac{π}{2}$ is a local minimum

$x = \frac{π}{6}$ is a local maximum

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

$x = \frac{3 π}{2}$ is a local maximum

Step 21