Calculus Examples

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

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

Differentiate.

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

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

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

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 $f (x) = \cos (x)$ and $g (x) = 2 x$ .

Tap for more steps...

Step 1.2.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

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

Step 1.2.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

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

Step 1.2.3

Replace all occurrences of $u_{1}$ with $2 x$ .

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

Step 1.3

Differentiate.

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

Multiply $2$ by $- 1$ .

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

Step 1.3.3

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

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

Step 1.3.4

Multiply $- 2$ by $1$ .

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

Step 1.3.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin (2 x)$ with respect to $x$ is $- \frac{d}{d x} [\sin (2 x)]$ .

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

Step 1.4

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 $g (x) = 2 x$ .

Tap for more steps...

Step 1.4.1

To apply the Chain Rule, set $u_{2}$ as $2 x$ .

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

Step 1.4.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

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

Step 1.4.3

Replace all occurrences of $u_{2}$ with $2 x$ .

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

Step 1.5

Differentiate.

Tap for more steps...

Step 1.5.1

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

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

Step 1.5.2

Multiply $2$ by $- 1$ .

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

Step 1.5.3

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

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

Step 1.5.4

Multiply $- 2$ by $1$ .

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

Step 1.6

Simplify.

Tap for more steps...

Step 1.6.1

Apply the distributive property.

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

Step 1.6.2

Combine terms.

Tap for more steps...

Step 1.6.2.1

Multiply $- 2$ by $2$ .

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

Step 1.6.2.2

Multiply $- 2$ by $2$ .

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

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 \sin (2 x)$ with respect to $x$ is $- 4 \frac{d}{d x} [\sin (2 x)]$ .

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

Step 2.2.2

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

Tap for more steps...

Step 2.2.2.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

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

Step 2.2.2.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

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

Step 2.2.2.3

Replace all occurrences of $u_{1}$ with $2 x$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $2$ by $1$ .

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

Step 2.2.6

Move $2$ to the left of $\cos (2 x)$ .

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

Step 2.2.7

Multiply $2$ by $- 4$ .

$f'' (x) = - 8 \cos (2 x) + \frac{d}{d x} (- 4 \cos (2 x))$

Step 2.3

Evaluate $\frac{d}{d x} [- 4 \cos (2 x)]$ .

Tap for more steps...

Step 2.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 \cos (2 x)$ with respect to $x$ is $- 4 \frac{d}{d x} [\cos (2 x)]$ .

$f'' (x) = - 8 \cos (2 x) - 4 \frac{d}{d x} \cos (2 x)$

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 $g (x) = 2 x$ .

Tap for more steps...

Step 2.3.2.1

To apply the Chain Rule, set $u_{2}$ as $2 x$ .

$f'' (x) = - 8 \cos (2 x) - 4 (\frac{d}{d u (2)} (\cos (u_{2})) \frac{d}{d x} (2 x))$

Step 2.3.2.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$f'' (x) = - 8 \cos (2 x) - 4 (- \sin (u_{2}) \frac{d}{d x} (2 x))$

Step 2.3.2.3

Replace all occurrences of $u_{2}$ with $2 x$ .

$f'' (x) = - 8 \cos (2 x) - 4 (- \sin (2 x) \frac{d}{d x} (2 x))$

Step 2.3.3

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

$f'' (x) = - 8 \cos (2 x) - 4 (- \sin (2 x) (2 \frac{d}{d x} (x)))$

Step 2.3.4

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

$f'' (x) = - 8 \cos (2 x) - 4 (- \sin (2 x) (2 \cdot 1))$

Step 2.3.5

Multiply $2$ by $1$ .

$f'' (x) = - 8 \cos (2 x) - 4 (- \sin (2 x) \cdot 2)$

Step 2.3.6

Multiply $2$ by $- 1$ .

$f'' (x) = - 8 \cos (2 x) - 4 (- 2 \sin (2 x))$

Step 2.3.7

Multiply $- 2$ by $- 4$ .

$f'' (x) = - 8 \cos (2 x) + 8 \sin (2 x)$

Step 3

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

$- 4 \sin (2 x) - 4 \cos (2 x) = 0$

Step 4

Divide each term in the equation by $\cos (2 x)$ .

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

Step 5

Separate fractions.

$\frac{- 4}{1} \cdot \frac{\sin (2 x)}{\cos (2 x)} + \frac{- 4 \cos (2 x)}{\cos (2 x)} = \frac{0}{\cos (2 x)}$

Step 6

Convert from $\frac{\sin (2 x)}{\cos (2 x)}$ to $\tan (2 x)$ .

$\frac{- 4}{1} \cdot \tan (2 x) + \frac{- 4 \cos (2 x)}{\cos (2 x)} = \frac{0}{\cos (2 x)}$

Step 7

Divide $- 4$ by $1$ .

$- 4 \tan (2 x) + \frac{- 4 \cos (2 x)}{\cos (2 x)} = \frac{0}{\cos (2 x)}$

Step 8

Cancel the common factor of $\cos (2 x)$ .

Tap for more steps...

Step 8.1

Cancel the common factor.

$- 4 \tan (2 x) + \frac{- 4 \cos (2 x)}{\cos (2 x)} = \frac{0}{\cos (2 x)}$

Step 8.2

Divide $- 4$ by $1$ .

$- 4 \tan (2 x) - 4 = \frac{0}{\cos (2 x)}$

Step 9

Separate fractions.

$- 4 \tan (2 x) - 4 = \frac{0}{1} \cdot \frac{1}{\cos (2 x)}$

Step 10

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

$- 4 \tan (2 x) - 4 = \frac{0}{1} \cdot \sec (2 x)$

Step 11

Divide $0$ by $1$ .

$- 4 \tan (2 x) - 4 = 0 \sec (2 x)$

Step 12

Multiply $0$ by $\sec (2 x)$ .

$- 4 \tan (2 x) - 4 = 0$

Step 13

Add $4$ to both sides of the equation.

$- 4 \tan (2 x) = 4$

Step 14

Divide each term in $- 4 \tan (2 x) = 4$ by $- 4$ and simplify.

Tap for more steps...

Step 14.1

Divide each term in $- 4 \tan (2 x) = 4$ by $- 4$ .

$\frac{- 4 \tan (2 x)}{- 4} = \frac{4}{- 4}$

Step 14.2

Simplify the left side.

Tap for more steps...

Step 14.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 14.2.1.1

Cancel the common factor.

$\frac{- 4 \tan (2 x)}{- 4} = \frac{4}{- 4}$

Step 14.2.1.2

Divide $\tan (2 x)$ by $1$ .

$\tan (2 x) = \frac{4}{- 4}$

Step 14.3

Simplify the right side.

Tap for more steps...

Step 14.3.1

Divide $4$ by $- 4$ .

$\tan (2 x) = - 1$

Step 15

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

$2 x = \arctan (- 1)$

Step 16

Simplify the right side.

Tap for more steps...

Step 16.1

The exact value of $\arctan (- 1)$ is $- \frac{π}{4}$ .

$2 x = - \frac{π}{4}$

Step 17

Divide each term in $2 x = - \frac{π}{4}$ by $2$ and simplify.

Tap for more steps...

Step 17.1

Divide each term in $2 x = - \frac{π}{4}$ by $2$ .

$\frac{2 x}{2} = \frac{- \frac{π}{4}}{2}$

Step 17.2

Simplify the left side.

Tap for more steps...

Step 17.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 17.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- \frac{π}{4}}{2}$

Step 17.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{4}}{2}$

Step 17.3

Simplify the right side.

Tap for more steps...

Step 17.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{4} \cdot \frac{1}{2}$

Step 17.3.2

Multiply $- \frac{π}{4} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 17.3.2.1

Multiply $\frac{1}{2}$ by $\frac{π}{4}$ .

$x = - \frac{π}{2 \cdot 4}$

Step 17.3.2.2

Multiply $2$ by $4$ .

$x = - \frac{π}{8}$

Step 18

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

$2 x = - \frac{π}{4} - π$

Step 19

Simplify the expression to find the second solution.

Tap for more steps...

Step 19.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$2 x = - \frac{π}{4} - π + 2 π$

Step 19.2

The resulting angle of $\frac{3 π}{4}$ is positive and coterminal with $- \frac{π}{4} - π$ .

$2 x = \frac{3 π}{4}$

Step 19.3

Divide each term in $2 x = \frac{3 π}{4}$ by $2$ and simplify.

Tap for more steps...

Step 19.3.1

Divide each term in $2 x = \frac{3 π}{4}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{3 π}{4}}{2}$

Step 19.3.2

Simplify the left side.

Tap for more steps...

Step 19.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 19.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{3 π}{4}}{2}$

Step 19.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{4}}{2}$

Step 19.3.3

Simplify the right side.

Tap for more steps...

Step 19.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{4} \cdot \frac{1}{2}$

Step 19.3.3.2

Multiply $\frac{3 π}{4} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 19.3.3.2.1

Multiply $\frac{3 π}{4}$ by $\frac{1}{2}$ .

$x = \frac{3 π}{4 \cdot 2}$

Step 19.3.3.2.2

Multiply $4$ by $2$ .

$x = \frac{3 π}{8}$

Step 20

The solution to the equation $2 x = - \frac{π}{4}$ .

$x = - \frac{π}{8}, \frac{3 π}{8}$

Step 21

Evaluate the second derivative at $x = - \frac{π}{8}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 8 \cos (2 (- \frac{π}{8})) + 8 \sin (2 (- \frac{π}{8}))$

Step 22

Evaluate the second derivative.

Tap for more steps...

Step 22.1

Simplify each term.

Tap for more steps...

Step 22.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 22.1.1.1

Move the leading negative in $- \frac{π}{8}$ into the numerator.

$- 8 \cos (2 \frac{- π}{8}) + 8 \sin (2 (- \frac{π}{8}))$

Step 22.1.1.2

Factor $2$ out of $8$ .

$- 8 \cos (2 \frac{- π}{2 (4)}) + 8 \sin (2 (- \frac{π}{8}))$

Step 22.1.1.3

Cancel the common factor.

$- 8 \cos (2 \frac{- π}{2 \cdot 4}) + 8 \sin (2 (- \frac{π}{8}))$

Step 22.1.1.4

Rewrite the expression.

$- 8 \cos (\frac{- π}{4}) + 8 \sin (2 (- \frac{π}{8}))$

Step 22.1.2

Move the negative in front of the fraction.

$- 8 \cos (- \frac{π}{4}) + 8 \sin (2 (- \frac{π}{8}))$

Step 22.1.3

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$- 8 \cos (\frac{7 π}{4}) + 8 \sin (2 (- \frac{π}{8}))$

Step 22.1.4

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

$- 8 \cos (\frac{π}{4}) + 8 \sin (2 (- \frac{π}{8}))$

Step 22.1.5

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- 8 \frac{\sqrt{2}}{2} + 8 \sin (2 (- \frac{π}{8}))$

Step 22.1.6

Cancel the common factor of $2$ .

Tap for more steps...

Step 22.1.6.1

Factor $2$ out of $- 8$ .

$2 (- 4) \frac{\sqrt{2}}{2} + 8 \sin (2 (- \frac{π}{8}))$

Step 22.1.6.2

Cancel the common factor.

$2 \cdot - 4 \frac{\sqrt{2}}{2} + 8 \sin (2 (- \frac{π}{8}))$

Step 22.1.6.3

Rewrite the expression.

$- 4 \sqrt{2} + 8 \sin (2 (- \frac{π}{8}))$

Step 22.1.7

Cancel the common factor of $2$ .

Tap for more steps...

Step 22.1.7.1

Move the leading negative in $- \frac{π}{8}$ into the numerator.

$- 4 \sqrt{2} + 8 \sin (2 \frac{- π}{8})$

Step 22.1.7.2

Factor $2$ out of $8$ .

$- 4 \sqrt{2} + 8 \sin (2 \frac{- π}{2 (4)})$

Step 22.1.7.3

Cancel the common factor.

$- 4 \sqrt{2} + 8 \sin (2 \frac{- π}{2 \cdot 4})$

Step 22.1.7.4

Rewrite the expression.

$- 4 \sqrt{2} + 8 \sin (\frac{- π}{4})$

Step 22.1.8

Move the negative in front of the fraction.

$- 4 \sqrt{2} + 8 \sin (- \frac{π}{4})$

Step 22.1.9

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$- 4 \sqrt{2} + 8 \sin (\frac{7 π}{4})$

Step 22.1.10

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.

$- 4 \sqrt{2} + 8 (- \sin (\frac{π}{4}))$

Step 22.1.11

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- 4 \sqrt{2} + 8 (- \frac{\sqrt{2}}{2})$

Step 22.1.12

Cancel the common factor of $2$ .

Tap for more steps...

Step 22.1.12.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

$- 4 \sqrt{2} + 8 \frac{- \sqrt{2}}{2}$

Step 22.1.12.2

Factor $2$ out of $8$ .

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

Step 22.1.12.3

Cancel the common factor.

$- 4 \sqrt{2} + 2 \cdot 4 \frac{- \sqrt{2}}{2}$

Step 22.1.12.4

Rewrite the expression.

$- 4 \sqrt{2} + 4 (- \sqrt{2})$

Step 22.1.13

Multiply $- 1$ by $4$ .

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

Step 22.2

Subtract $4 \sqrt{2}$ from $- 4 \sqrt{2}$ .

$- 8 \sqrt{2}$

Step 23

$x = - \frac{π}{8}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - \frac{π}{8}$ is a local maximum

Step 24

Find the y-value when $x = - \frac{π}{8}$ .

Tap for more steps...

Step 24.1

Replace the variable $x$ with $- \frac{π}{8}$ in the expression.

$f (- \frac{π}{8}) = 2 (\cos (2 (- \frac{π}{8})) - \sin (2 (- \frac{π}{8})))$

Step 24.2

Simplify the result.

Tap for more steps...

Step 24.2.1

Simplify each term.

Tap for more steps...

Step 24.2.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 24.2.1.1.1

Move the leading negative in $- \frac{π}{8}$ into the numerator.

$f (- \frac{π}{8}) = 2 (\cos (2 (\frac{- π}{8})) - \sin (2 (- \frac{π}{8})))$

Step 24.2.1.1.2

Factor $2$ out of $8$ .

$f (- \frac{π}{8}) = 2 (\cos (2 (\frac{- π}{2 (4)})) - \sin (2 (- \frac{π}{8})))$

Step 24.2.1.1.3

Cancel the common factor.

$f (- \frac{π}{8}) = 2 (\cos (2 (\frac{- π}{2 \cdot 4})) - \sin (2 (- \frac{π}{8})))$

Step 24.2.1.1.4

Rewrite the expression.

$f (- \frac{π}{8}) = 2 (\cos (\frac{- π}{4}) - \sin (2 (- \frac{π}{8})))$

Step 24.2.1.2

Move the negative in front of the fraction.

$f (- \frac{π}{8}) = 2 (\cos (- \frac{π}{4}) - \sin (2 (- \frac{π}{8})))$

Step 24.2.1.3

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (- \frac{π}{8}) = 2 (\cos (\frac{7 π}{4}) - \sin (2 (- \frac{π}{8})))$

Step 24.2.1.4

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

$f (- \frac{π}{8}) = 2 (\cos (\frac{π}{4}) - \sin (2 (- \frac{π}{8})))$

Step 24.2.1.5

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$f (- \frac{π}{8}) = 2 (\frac{\sqrt{2}}{2} - \sin (2 (- \frac{π}{8})))$

Step 24.2.1.6

Cancel the common factor of $2$ .

Tap for more steps...

Step 24.2.1.6.1

Move the leading negative in $- \frac{π}{8}$ into the numerator.

$f (- \frac{π}{8}) = 2 (\frac{\sqrt{2}}{2} - \sin (2 (\frac{- π}{8})))$

Step 24.2.1.6.2

Factor $2$ out of $8$ .

$f (- \frac{π}{8}) = 2 (\frac{\sqrt{2}}{2} - \sin (2 (\frac{- π}{2 (4)})))$

Step 24.2.1.6.3

Cancel the common factor.

$f (- \frac{π}{8}) = 2 (\frac{\sqrt{2}}{2} - \sin (2 (\frac{- π}{2 \cdot 4})))$

Step 24.2.1.6.4

Rewrite the expression.

$f (- \frac{π}{8}) = 2 (\frac{\sqrt{2}}{2} - \sin (\frac{- π}{4}))$

Step 24.2.1.7

Move the negative in front of the fraction.

$f (- \frac{π}{8}) = 2 (\frac{\sqrt{2}}{2} - \sin (- \frac{π}{4}))$

Step 24.2.1.8

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (- \frac{π}{8}) = 2 (\frac{\sqrt{2}}{2} - \sin (\frac{7 π}{4}))$

Step 24.2.1.9

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.

$f (- \frac{π}{8}) = 2 (\frac{\sqrt{2}}{2} + \sin (\frac{π}{4}))$

Step 24.2.1.10

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$f (- \frac{π}{8}) = 2 (\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2})$

Step 24.2.1.11

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

Tap for more steps...

Step 24.2.1.11.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{π}{8}) = 2 (\frac{\sqrt{2}}{2} + 1 (\frac{\sqrt{2}}{2}))$

Step 24.2.1.11.2

Multiply $\frac{\sqrt{2}}{2}$ by $1$ .

$f (- \frac{π}{8}) = 2 (\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2})$

Step 24.2.2

Simplify terms.

Tap for more steps...

Step 24.2.2.1

Combine the numerators over the common denominator.

$f (- \frac{π}{8}) = 2 (\frac{\sqrt{2} + \sqrt{2}}{2})$

Step 24.2.2.2

Add $\sqrt{2}$ and $\sqrt{2}$ .

$f (- \frac{π}{8}) = 2 (\frac{2 \sqrt{2}}{2})$

Step 24.2.2.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 24.2.2.3.1

Cancel the common factor.

$f (- \frac{π}{8}) = 2 (\frac{2 \sqrt{2}}{2})$

Step 24.2.2.3.2

Rewrite the expression.

$f (- \frac{π}{8}) = 2 \sqrt{2}$

Step 24.2.3

The final answer is $2 \sqrt{2}$ .

$y = 2 \sqrt{2}$

Step 25

Evaluate the second derivative at $x = \frac{3 π}{8}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 8 \cos (2 (\frac{3 π}{8})) + 8 \sin (2 (\frac{3 π}{8}))$

Step 26

Evaluate the second derivative.

Tap for more steps...

Step 26.1

Simplify each term.

Tap for more steps...

Step 26.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 26.1.1.1

Factor $2$ out of $8$ .

$- 8 \cos (2 \frac{3 π}{2 (4)}) + 8 \sin (2 (\frac{3 π}{8}))$

Step 26.1.1.2

Cancel the common factor.

$- 8 \cos (2 \frac{3 π}{2 \cdot 4}) + 8 \sin (2 (\frac{3 π}{8}))$

Step 26.1.1.3

Rewrite the expression.

$- 8 \cos (\frac{3 π}{4}) + 8 \sin (2 (\frac{3 π}{8}))$

Step 26.1.2

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.

$- 8 (- \cos (\frac{π}{4})) + 8 \sin (2 (\frac{3 π}{8}))$

Step 26.1.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- 8 (- \frac{\sqrt{2}}{2}) + 8 \sin (2 (\frac{3 π}{8}))$

Step 26.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 26.1.4.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

$- 8 \frac{- \sqrt{2}}{2} + 8 \sin (2 (\frac{3 π}{8}))$

Step 26.1.4.2

Factor $2$ out of $- 8$ .

$2 (- 4) \frac{- \sqrt{2}}{2} + 8 \sin (2 (\frac{3 π}{8}))$

Step 26.1.4.3

Cancel the common factor.

$2 \cdot - 4 \frac{- \sqrt{2}}{2} + 8 \sin (2 (\frac{3 π}{8}))$

Step 26.1.4.4

Rewrite the expression.

$- 4 (- \sqrt{2}) + 8 \sin (2 (\frac{3 π}{8}))$

Step 26.1.5

Multiply $- 1$ by $- 4$ .

$4 \sqrt{2} + 8 \sin (2 (\frac{3 π}{8}))$

Step 26.1.6

Cancel the common factor of $2$ .

Tap for more steps...

Step 26.1.6.1

Factor $2$ out of $8$ .

$4 \sqrt{2} + 8 \sin (2 \frac{3 π}{2 (4)})$

Step 26.1.6.2

Cancel the common factor.

$4 \sqrt{2} + 8 \sin (2 \frac{3 π}{2 \cdot 4})$

Step 26.1.6.3

Rewrite the expression.

$4 \sqrt{2} + 8 \sin (\frac{3 π}{4})$

Step 26.1.7

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

$4 \sqrt{2} + 8 \sin (\frac{π}{4})$

Step 26.1.8

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$4 \sqrt{2} + 8 \frac{\sqrt{2}}{2}$

Step 26.1.9

Cancel the common factor of $2$ .

Tap for more steps...

Step 26.1.9.1

Factor $2$ out of $8$ .

$4 \sqrt{2} + 2 (4) \frac{\sqrt{2}}{2}$

Step 26.1.9.2

Cancel the common factor.

$4 \sqrt{2} + 2 \cdot 4 \frac{\sqrt{2}}{2}$

Step 26.1.9.3

Rewrite the expression.

$4 \sqrt{2} + 4 \sqrt{2}$

Step 26.2

Add $4 \sqrt{2}$ and $4 \sqrt{2}$ .

$8 \sqrt{2}$

Step 27

$x = \frac{3 π}{8}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{3 π}{8}$ is a local minimum

Step 28

Find the y-value when $x = \frac{3 π}{8}$ .

Tap for more steps...

Step 28.1

Replace the variable $x$ with $\frac{3 π}{8}$ in the expression.

$f (\frac{3 π}{8}) = 2 (\cos (2 (\frac{3 π}{8})) - \sin (2 (\frac{3 π}{8})))$

Step 28.2

Simplify the result.

Tap for more steps...

Step 28.2.1

Simplify each term.

Tap for more steps...

Step 28.2.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 28.2.1.1.1

Factor $2$ out of $8$ .

$f (\frac{3 π}{8}) = 2 (\cos (2 (\frac{3 π}{2 (4)})) - \sin (2 (\frac{3 π}{8})))$

Step 28.2.1.1.2

Cancel the common factor.

$f (\frac{3 π}{8}) = 2 (\cos (2 (\frac{3 π}{2 \cdot 4})) - \sin (2 (\frac{3 π}{8})))$

Step 28.2.1.1.3

Rewrite the expression.

$f (\frac{3 π}{8}) = 2 (\cos (\frac{3 π}{4}) - \sin (2 (\frac{3 π}{8})))$

Step 28.2.1.2

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.

$f (\frac{3 π}{8}) = 2 (- \cos (\frac{π}{4}) - \sin (2 (\frac{3 π}{8})))$

Step 28.2.1.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$f (\frac{3 π}{8}) = 2 (- \frac{\sqrt{2}}{2} - \sin (2 (\frac{3 π}{8})))$

Step 28.2.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 28.2.1.4.1

Factor $2$ out of $8$ .

$f (\frac{3 π}{8}) = 2 (- \frac{\sqrt{2}}{2} - \sin (2 (\frac{3 π}{2 (4)})))$

Step 28.2.1.4.2

Cancel the common factor.

$f (\frac{3 π}{8}) = 2 (- \frac{\sqrt{2}}{2} - \sin (2 (\frac{3 π}{2 \cdot 4})))$

Step 28.2.1.4.3

Rewrite the expression.

$f (\frac{3 π}{8}) = 2 (- \frac{\sqrt{2}}{2} - \sin (\frac{3 π}{4}))$

Step 28.2.1.5

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

$f (\frac{3 π}{8}) = 2 (- \frac{\sqrt{2}}{2} - \sin (\frac{π}{4}))$

Step 28.2.1.6

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$f (\frac{3 π}{8}) = 2 (- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2})$

Step 28.2.2

Simplify terms.

Tap for more steps...

Step 28.2.2.1

Combine the numerators over the common denominator.

$f (\frac{3 π}{8}) = 2 (\frac{- \sqrt{2} - \sqrt{2}}{2})$

Step 28.2.2.2

Subtract $\sqrt{2}$ from $- \sqrt{2}$ .

$f (\frac{3 π}{8}) = 2 (\frac{- 2 \sqrt{2}}{2})$

Step 28.2.2.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 28.2.2.3.1

Cancel the common factor.

$f (\frac{3 π}{8}) = 2 (\frac{- 2 \sqrt{2}}{2})$

Step 28.2.2.3.2

Rewrite the expression.

$f (\frac{3 π}{8}) = - 2 \sqrt{2}$

Step 28.2.3

The final answer is $- 2 \sqrt{2}$ .

$y = - 2 \sqrt{2}$

Step 29

These are the local extrema for $f (x) = 2 (\cos (2 x) - \sin (2 x))$ .

$(- \frac{π}{8}, 2 \sqrt{2})$ is a local maxima

$(\frac{3 π}{8}, - 2 \sqrt{2})$ is a local minima

Step 30