Find the Maximum/Minimum Value y=-3+4cos((5pi)/6(x+4))

$y = - 3 + 4 \cos (\frac{5 π}{6} (x + 4))$

Step 1

Find the first derivative of the function.

Differentiate.

By the Sum Rule, the derivative of $- 3 + 4 \cos (\frac{5 π}{6} \cdot (x + 4))$ with respect to $x$ is $\frac{d}{d x} [- 3] + \frac{d}{d x} [4 \cos (\frac{5 π}{6} \cdot (x + 4))]$ .

$\frac{d}{d x} [- 3] + \frac{d}{d x} [4 \cos (\frac{5 π}{6} \cdot (x + 4))]$

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [4 \cos (\frac{5 π}{6} \cdot (x + 4))]$

Evaluate $\frac{d}{d x} [4 \cos (\frac{5 π}{6} \cdot (x + 4))]$ .

Since $4$ is constant with respect to $x$ , the derivative of $4 \cos (\frac{5 π}{6} (x + 4))$ with respect to $x$ is $4 \frac{d}{d x} [\cos (\frac{5 π}{6} (x + 4))]$ .

$0 + 4 \frac{d}{d x} [\cos (\frac{5 π}{6} (x + 4))]$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = \frac{5 π}{6} (x + 4)$ .

To apply the Chain Rule, set $u$ as $\frac{5 π}{6} (x + 4)$ .

$0 + 4 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [\frac{5 π}{6} (x + 4)])$

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

$0 + 4 (- \sin (\frac{5 π}{6} (x + 4)) \frac{d}{d x} [\frac{5 π}{6} (x + 4)])$

Since $\frac{5 π}{6}$ is constant with respect to $x$ , the derivative of $\frac{5 π}{6} (x + 4)$ with respect to $x$ is $\frac{5 π}{6} \frac{d}{d x} [x + 4]$ .

$0 + 4 (- \sin (\frac{5 π}{6} (x + 4)) (\frac{5 π}{6} \frac{d}{d x} [x + 4]))$

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

$0 + 4 (- \sin (\frac{5 π}{6} (x + 4)) (\frac{5 π}{6} (\frac{d}{d x} [x] + \frac{d}{d x} [4])))$

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

$0 + 4 (- \sin (\frac{5 π}{6} (x + 4)) (\frac{5 π}{6} (1 + \frac{d}{d x} [4])))$

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + 4 (- \sin (\frac{5 π}{6} (x + 4)) (\frac{5 π}{6} (1 + 0)))$

Add $1$ and $0$ .

$0 + 4 (- \sin (\frac{5 π}{6} (x + 4)) (\frac{5 π}{6} \cdot 1))$

Multiply $\frac{5 π}{6}$ by $1$ .

$0 + 4 (- \sin (\frac{5 π}{6} (x + 4)) \frac{5 π}{6})$

Combine $\frac{5 π}{6}$ and $\sin (\frac{5 π}{6} (x + 4))$ .

$0 + 4 (- \frac{5 π \sin (\frac{5 π}{6} (x + 4))}{6})$

Multiply $- 1$ by $4$ .

$0 - 4 \frac{5 π \sin (\frac{5 π}{6} (x + 4))}{6}$

Combine $- 4$ and $\frac{5 π \sin (\frac{5 π}{6} (x + 4))}{6}$ .

$0 + \frac{- 4 (5 π \sin (\frac{5 π}{6} (x + 4)))}{6}$

Multiply $5$ by $- 4$ .

$0 + \frac{- 20 (π \sin (\frac{5 π}{6} (x + 4)))}{6}$

Cancel the common factor of $- 20$ and $6$ .

Factor $2$ out of $- 20 π \sin (\frac{5 π}{6} (x + 4))$ .

$0 + \frac{2 (- 10 π \sin (\frac{5 π}{6} (x + 4)))}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$0 + \frac{2 (- 10 π \sin (\frac{5 π}{6} (x + 4)))}{2 (3)}$

Cancel the common factor.

$0 + \frac{2 (- 10 π \sin (\frac{5 π}{6} (x + 4)))}{2 \cdot 3}$

Rewrite the expression.

$0 + \frac{- 10 π \sin (\frac{5 π}{6} (x + 4))}{3}$

Move the negative in front of the fraction.

$0 - \frac{10 π \sin (\frac{5 π}{6} (x + 4))}{3}$

Simplify.

Apply the distributive property.

$0 - \frac{10 π \sin (\frac{5 π}{6} x + \frac{5 π}{6} \cdot 4)}{3}$

Combine terms.

Combine $\frac{5 π}{6}$ and $4$ .

$0 - \frac{10 π \sin (\frac{5 π}{6} x + \frac{5 π \cdot 4}{6})}{3}$

Multiply $4$ by $5$ .

$0 - \frac{10 π \sin (\frac{5 π}{6} x + \frac{20 π}{6})}{3}$

Cancel the common factor of $20$ and $6$ .

Factor $2$ out of $20 π$ .

$0 - \frac{10 π \sin (\frac{5 π}{6} x + \frac{2 (10 π)}{6})}{3}$

Cancel the common factors.

Factor $2$ out of $6$ .

$0 - \frac{10 π \sin (\frac{5 π}{6} x + \frac{2 (10 π)}{2 (3)})}{3}$

Cancel the common factor.

$0 - \frac{10 π \sin (\frac{5 π}{6} x + \frac{2 (10 π)}{2 \cdot 3})}{3}$

Rewrite the expression.

$0 - \frac{10 π \sin (\frac{5 π}{6} x + \frac{10 π}{3})}{3}$

Combine $\frac{5 π}{6}$ and $x$ .

$0 - \frac{10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{3}$

Subtract $\frac{10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{3}$ from $0$ .

$- \frac{10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{3}$

Step 2

Find the second derivative of the function.

Since $- \frac{10 π}{3}$ is constant with respect to $x$ , the derivative of $- \frac{10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{3}$ with respect to $x$ is $- \frac{10 π}{3} \frac{d}{d x} [\sin (\frac{5 π x}{6} + \frac{10 π}{3})]$ .

$f'' (x) = - \frac{10 π}{3} \cdot \frac{d}{d x} \sin (\frac{5 π x}{6} + \frac{10 π}{3})$

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 $g (x) = \frac{5 π x}{6} + \frac{10 π}{3}$ .

To apply the Chain Rule, set $u$ as $\frac{5 π x}{6} + \frac{10 π}{3}$ .

$f'' (x) = - \frac{10 π}{3} \cdot (\frac{d}{d u} (\sin (u)) \frac{d}{d x} (\frac{5 π x}{6} + \frac{10 π}{3}))$

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

$f'' (x) = - \frac{10 π}{3} \cdot (\cos (u) \frac{d}{d x} (\frac{5 π x}{6} + \frac{10 π}{3}))$

Replace all occurrences of $u$ with $\frac{5 π x}{6} + \frac{10 π}{3}$ .

$f'' (x) = - \frac{10 π}{3} \cdot (\cos (\frac{5 π x}{6} + \frac{10 π}{3}) \frac{d}{d x} (\frac{5 π x}{6} + \frac{10 π}{3}))$

Differentiate.

Combine $\cos (\frac{5 π x}{6} + \frac{10 π}{3})$ and $\frac{10 π}{3}$ .

$f'' (x) = - \frac{\cos (\frac{5 π x}{6} + \frac{10 π}{3}) (10 π)}{3} \cdot \frac{d}{d x} (\frac{5 π x}{6} + \frac{10 π}{3})$

Move $10$ to the left of $\cos (\frac{5 π x}{6} + \frac{10 π}{3})$ .

$f'' (x) = - \frac{10 \cdot (\cos (\frac{5 π x}{6} + \frac{10 π}{3}) π)}{3} \cdot \frac{d}{d x} (\frac{5 π x}{6} + \frac{10 π}{3})$

By the Sum Rule, the derivative of $\frac{5 π x}{6} + \frac{10 π}{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{5 π x}{6}] + \frac{d}{d x} [\frac{10 π}{3}]$ .

$f'' (x) = - \frac{10 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) π}{3} \cdot (\frac{d}{d x} (\frac{5 π x}{6}) + \frac{d}{d x} (\frac{10 π}{3}))$

Since $\frac{5 π}{6}$ is constant with respect to $x$ , the derivative of $\frac{5 π x}{6}$ with respect to $x$ is $\frac{5 π}{6} \frac{d}{d x} [x]$ .

$f'' (x) = - \frac{10 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) π}{3} \cdot (\frac{5 π}{6} \cdot \frac{d}{d x} (x) + \frac{d}{d x} (\frac{10 π}{3}))$

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

$f'' (x) = - \frac{10 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) π}{3} \cdot (\frac{5 π}{6} \cdot 1 + \frac{d}{d x} (\frac{10 π}{3}))$

Multiply $\frac{5 π}{6}$ by $1$ .

$f'' (x) = - \frac{10 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) π}{3} \cdot (\frac{5 π}{6} + \frac{d}{d x} (\frac{10 π}{3}))$

Since $\frac{10 π}{3}$ is constant with respect to $x$ , the derivative of $\frac{10 π}{3}$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{10 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) π}{3} \cdot (\frac{5 π}{6} + 0)$

Combine fractions.

Add $\frac{5 π}{6}$ and $0$ .

$f'' (x) = - \frac{10 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) π}{3} \cdot \frac{5 π}{6}$

Multiply $\frac{5 π}{6}$ by $\frac{10 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) π}{3}$ .

$f'' (x) = - \frac{5 π (10 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) π)}{6 \cdot 3}$

Multiply $10$ by $5$ .

$f'' (x) = - \frac{50 π (\cos (\frac{5 π x}{6} + \frac{10 π}{3}) π)}{6 \cdot 3}$

Raise $π$ to the power of $1$ .

$f'' (x) = - \frac{50 (π π) \cos (\frac{5 π x}{6} + \frac{10 π}{3})}{6 \cdot 3}$

Raise $π$ to the power of $1$ .

$f'' (x) = - \frac{50 (π π) \cos (\frac{5 π x}{6} + \frac{10 π}{3})}{6 \cdot 3}$

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

$f'' (x) = - \frac{50 π^{1 + 1} \cos (\frac{5 π x}{6} + \frac{10 π}{3})}{6 \cdot 3}$

Simplify the expression.

Add $1$ and $1$ .

$f'' (x) = - \frac{50 π^{2} \cos (\frac{5 π x}{6} + \frac{10 π}{3})}{6 \cdot 3}$

Multiply $6$ by $3$ .

$f'' (x) = - \frac{50 π^{2} \cos (\frac{5 π x}{6} + \frac{10 π}{3})}{18}$

Cancel the common factor of $50$ and $18$ .

Factor $2$ out of $50 π^{2} \cos (\frac{5 π x}{6} + \frac{10 π}{3})$ .

$f'' (x) = - \frac{2 (25 π^{2} \cos (\frac{5 π x}{6} + \frac{10 π}{3}))}{18}$

Cancel the common factors.

Factor $2$ out of $18$ .

$f'' (x) = - \frac{2 (25 π^{2} \cos (\frac{5 π x}{6} + \frac{10 π}{3}))}{2 (9)}$

Cancel the common factor.

$f'' (x) = - \frac{2 (25 π^{2} \cos (\frac{5 π x}{6} + \frac{10 π}{3}))}{2 \cdot 9}$

Rewrite the expression.

$f'' (x) = - \frac{25 π^{2} \cos (\frac{5 π x}{6} + \frac{10 π}{3})}{9}$

Step 3

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

$- \frac{10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{3} = 0$

Step 4

Set the numerator equal to zero.

$10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3}) = 0$

Step 5

Solve the equation for $x$ .

Divide each term in $10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3}) = 0$ by $10 π$ and simplify.

Divide each term in $10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3}) = 0$ by $10 π$ .

$\frac{10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{10 π} = \frac{0}{10 π}$

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{10 π} = \frac{0}{10 π}$

Rewrite the expression.

$\frac{π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{π} = \frac{0}{10 π}$

Cancel the common factor of $π$ .

Cancel the common factor.

$\frac{π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{π} = \frac{0}{10 π}$

Divide $\sin (\frac{5 π x}{6} + \frac{10 π}{3})$ by $1$ .

$\sin (\frac{5 π x}{6} + \frac{10 π}{3}) = \frac{0}{10 π}$

Simplify the right side.

Cancel the common factor of $0$ and $10$ .

Factor $10$ out of $0$ .

$\sin (\frac{5 π x}{6} + \frac{10 π}{3}) = \frac{10 (0)}{10 π}$

Cancel the common factors.

Factor $10$ out of $10 π$ .

$\sin (\frac{5 π x}{6} + \frac{10 π}{3}) = \frac{10 (0)}{10 (π)}$

Cancel the common factor.

$\sin (\frac{5 π x}{6} + \frac{10 π}{3}) = \frac{10 \cdot 0}{10 π}$

Rewrite the expression.

$\sin (\frac{5 π x}{6} + \frac{10 π}{3}) = \frac{0}{π}$

Divide $0$ by $π$ .

$\sin (\frac{5 π x}{6} + \frac{10 π}{3}) = 0$

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

$\frac{5 π x}{6} + \frac{10 π}{3} = \arcsin (0)$

Simplify the right side.

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

$\frac{5 π x}{6} + \frac{10 π}{3} = 0$

Subtract $\frac{10 π}{3}$ from both sides of the equation.

$\frac{5 π x}{6} = - \frac{10 π}{3}$

Multiply both sides of the equation by $\frac{6}{5 π}$ .

$\frac{6}{5 π} \cdot \frac{5 π x}{6} = \frac{6}{5 π} (- \frac{10 π}{3})$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{6}{5 π} \cdot \frac{5 π x}{6}$ .

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6}{5 π} \cdot \frac{5 π x}{6} = \frac{6}{5 π} (- \frac{10 π}{3})$

Rewrite the expression.

$\frac{1}{5 π} (5 π x) = \frac{6}{5 π} (- \frac{10 π}{3})$

Cancel the common factor of $5 π$ .

Factor $5 π$ out of $5 π x$ .

$\frac{1}{5 π} (5 π (x)) = \frac{6}{5 π} (- \frac{10 π}{3})$

Cancel the common factor.

$\frac{1}{5 π} (5 π x) = \frac{6}{5 π} (- \frac{10 π}{3})$

Rewrite the expression.

$x = \frac{6}{5 π} (- \frac{10 π}{3})$

Simplify the right side.

Simplify $\frac{6}{5 π} (- \frac{10 π}{3})$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{10 π}{3}$ into the numerator.

$x = \frac{6}{5 π} \cdot \frac{- 10 π}{3}$

Factor $3$ out of $6$ .

$x = \frac{3 (2)}{5 π} \cdot \frac{- 10 π}{3}$

Cancel the common factor.

$x = \frac{3 \cdot 2}{5 π} \cdot \frac{- 10 π}{3}$

Rewrite the expression.

$x = \frac{2}{5 π} (- 10 π)$

Cancel the common factor of $5 π$ .

Factor $5 π$ out of $- 10 π$ .

$x = \frac{2}{5 π} (5 π (- 2))$

Cancel the common factor.

$x = \frac{2}{5 π} (5 π \cdot - 2)$

Rewrite the expression.

$x = 2 \cdot - 2$

Multiply $2$ by $- 2$ .

$x = - 4$

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.

$\frac{5 π x}{6} + \frac{10 π}{3} = π - 0$

Solve for $x$ .

Subtract $0$ from $π$ .

$\frac{5 π x}{6} + \frac{10 π}{3} = π$

Move all terms not containing $x$ to the right side of the equation.

Subtract $\frac{10 π}{3}$ from both sides of the equation.

$\frac{5 π x}{6} = π - \frac{10 π}{3}$

To write $π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5 π x}{6} = π \cdot \frac{3}{3} - \frac{10 π}{3}$

Combine $π$ and $\frac{3}{3}$ .

$\frac{5 π x}{6} = \frac{π \cdot 3}{3} - \frac{10 π}{3}$

Combine the numerators over the common denominator.

$\frac{5 π x}{6} = \frac{π \cdot 3 - 10 π}{3}$

Simplify the numerator.

Move $3$ to the left of $π$ .

$\frac{5 π x}{6} = \frac{3 \cdot π - 10 π}{3}$

Subtract $10 π$ from $3 π$ .

$\frac{5 π x}{6} = \frac{- 7 π}{3}$

Move the negative in front of the fraction.

$\frac{5 π x}{6} = - \frac{7 π}{3}$

Multiply both sides of the equation by $\frac{6}{5 π}$ .

$\frac{6}{5 π} \cdot \frac{5 π x}{6} = \frac{6}{5 π} (- \frac{7 π}{3})$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{6}{5 π} \cdot \frac{5 π x}{6}$ .

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6}{5 π} \cdot \frac{5 π x}{6} = \frac{6}{5 π} (- \frac{7 π}{3})$

Rewrite the expression.

$\frac{1}{5 π} (5 π x) = \frac{6}{5 π} (- \frac{7 π}{3})$

Cancel the common factor of $5 π$ .

Factor $5 π$ out of $5 π x$ .

$\frac{1}{5 π} (5 π (x)) = \frac{6}{5 π} (- \frac{7 π}{3})$

Cancel the common factor.

$\frac{1}{5 π} (5 π x) = \frac{6}{5 π} (- \frac{7 π}{3})$

Rewrite the expression.

$x = \frac{6}{5 π} (- \frac{7 π}{3})$

Simplify the right side.

Simplify $\frac{6}{5 π} (- \frac{7 π}{3})$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{7 π}{3}$ into the numerator.

$x = \frac{6}{5 π} \cdot \frac{- 7 π}{3}$

Factor $3$ out of $6$ .

$x = \frac{3 (2)}{5 π} \cdot \frac{- 7 π}{3}$

Cancel the common factor.

$x = \frac{3 \cdot 2}{5 π} \cdot \frac{- 7 π}{3}$

Rewrite the expression.

$x = \frac{2}{5 π} (- 7 π)$

Cancel the common factor of $π$ .

Factor $π$ out of $5 π$ .

$x = \frac{2}{π \cdot 5} (- 7 π)$

Factor $π$ out of $- 7 π$ .

$x = \frac{2}{π \cdot 5} (π \cdot - 7)$

Cancel the common factor.

$x = \frac{2}{π \cdot 5} (π \cdot - 7)$

Rewrite the expression.

$x = \frac{2}{5} \cdot - 7$

Combine $\frac{2}{5}$ and $- 7$ .

$x = \frac{2 \cdot - 7}{5}$

Simplify the expression.

Multiply $2$ by $- 7$ .

$x = \frac{- 14}{5}$

Move the negative in front of the fraction.

$x = - \frac{14}{5}$

The solution to the equation $\frac{5 π x}{6} + \frac{10 π}{3} = 0$ .

$x = - 4, - \frac{14}{5}$

Step 6

Evaluate the second derivative at $x = - 4$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{25 π^{2} \cos (\frac{5 π (- 4)}{6} + \frac{10 π}{3})}{9}$

Step 7

Evaluate the second derivative.

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 4$ and $6$ .

Factor $2$ out of $5 π (- 4)$ .

$- \frac{25 π^{2} \cos (\frac{2 (5 π \cdot (- 2))}{6} + \frac{10 π}{3})}{9}$

Cancel the common factors.

Factor $2$ out of $6$ .

$- \frac{25 π^{2} \cos (\frac{2 (5 π \cdot (- 2))}{2 (3)} + \frac{10 π}{3})}{9}$

Cancel the common factor.

$- \frac{25 π^{2} \cos (\frac{2 (5 π \cdot (- 2))}{2 \cdot 3} + \frac{10 π}{3})}{9}$

Rewrite the expression.

$- \frac{25 π^{2} \cos (\frac{5 π \cdot (- 2)}{3} + \frac{10 π}{3})}{9}$

Multiply $- 2$ by $5$ .

$- \frac{25 π^{2} \cos (\frac{- 10 π}{3} + \frac{10 π}{3})}{9}$

Simplify the numerator.

Combine the numerators over the common denominator.

$- \frac{25 π^{2} \cos (\frac{- 10 π + 10 π}{3})}{9}$

Add $- 10 π$ and $10 π$ .

$- \frac{25 π^{2} \cos (\frac{0}{3})}{9}$

Divide $0$ by $3$ .

$- \frac{25 π^{2} \cos (0)}{9}$

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

$- \frac{25 π^{2} \cdot 1}{9}$

Multiply $25$ by $1$ .

$- \frac{25 π^{2}}{9}$

Step 8

$x = - 4$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - 4$ is a local maximum

Step 9

Find the y-value when $x = - 4$ .

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

$f (- 4) = - 3 + 4 \cos (\frac{5 π}{6} \cdot ((- 4) + 4))$

Simplify the result.

Simplify each term.

Add $- 4$ and $4$ .

$f (- 4) = - 3 + 4 \cos (\frac{5 π}{6} \cdot 0)$

Multiply $\frac{5 π}{6}$ by $0$ .

$f (- 4) = - 3 + 4 \cos (0)$

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

$f (- 4) = - 3 + 4 \cdot 1$

Multiply $4$ by $1$ .

$f (- 4) = - 3 + 4$

Add $- 3$ and $4$ .

$f (- 4) = 1$

The final answer is $1$ .

$y = 1$

Step 10

Evaluate the second derivative at $x = - \frac{14}{5}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{25 π^{2} \cos (\frac{5 π (- \frac{14}{5})}{6} + \frac{10 π}{3})}{9}$

Step 11

Evaluate the second derivative.

Simplify the numerator.

Multiply $- 1$ by $5$ .

$- \frac{25 π^{2} \cos (\frac{- 5 π \frac{14}{5}}{6} + \frac{10 π}{3})}{9}$

Combine $\frac{14}{5}$ and $- 5$ .

$- \frac{25 π^{2} \cos (\frac{\frac{14 \cdot - 5}{5} π}{6} + \frac{10 π}{3})}{9}$

Combine $\frac{14 \cdot - 5}{5}$ and $π$ .

$- \frac{25 π^{2} \cos (\frac{\frac{14 \cdot - 5 π}{5}}{6} + \frac{10 π}{3})}{9}$

Multiply $14$ by $- 5$ .

$- \frac{25 π^{2} \cos (\frac{\frac{- 70 π}{5}}{6} + \frac{10 π}{3})}{9}$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{- 70 π}{5}$ by cancelling the common factors.

Factor $5$ out of $- 70 π$ .

$- \frac{25 π^{2} \cos (\frac{\frac{5 (- 14 π)}{5}}{6} + \frac{10 π}{3})}{9}$

Factor $5$ out of $5$ .

$- \frac{25 π^{2} \cos (\frac{\frac{5 (- 14 π)}{5 (1)}}{6} + \frac{10 π}{3})}{9}$

Cancel the common factor.

$- \frac{25 π^{2} \cos (\frac{\frac{5 (- 14 π)}{5 \cdot 1}}{6} + \frac{10 π}{3})}{9}$

Rewrite the expression.

$- \frac{25 π^{2} \cos (\frac{\frac{- 14 π}{1}}{6} + \frac{10 π}{3})}{9}$

Divide $- 14 π$ by $1$ .

$- \frac{25 π^{2} \cos (\frac{- 14 π}{6} + \frac{10 π}{3})}{9}$

Simplify the numerator.

Simplify each term.

Cancel the common factor of $- 14$ and $6$ .

Factor $2$ out of $- 14 π$ .

$- \frac{25 π^{2} \cos (\frac{2 (- 7 π)}{6} + \frac{10 π}{3})}{9}$

Cancel the common factors.

Factor $2$ out of $6$ .

$- \frac{25 π^{2} \cos (\frac{2 (- 7 π)}{2 (3)} + \frac{10 π}{3})}{9}$

Cancel the common factor.

$- \frac{25 π^{2} \cos (\frac{2 (- 7 π)}{2 \cdot 3} + \frac{10 π}{3})}{9}$

Rewrite the expression.

$- \frac{25 π^{2} \cos (\frac{- 7 π}{3} + \frac{10 π}{3})}{9}$

Move the negative in front of the fraction.

$- \frac{25 π^{2} \cos (- \frac{7 π}{3} + \frac{10 π}{3})}{9}$

Combine the numerators over the common denominator.

$- \frac{25 π^{2} \cos (\frac{- 7 π + 10 π}{3})}{9}$

Add $- 7 π$ and $10 π$ .

$- \frac{25 π^{2} \cos (\frac{3 π}{3})}{9}$

Cancel the common factor of $3$ .

Cancel the common factor.

$- \frac{25 π^{2} \cos (\frac{3 π}{3})}{9}$

Divide $π$ by $1$ .

$- \frac{25 π^{2} \cos (π)}{9}$

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.

$- \frac{25 π^{2} (- \cos (0))}{9}$

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

$- \frac{25 π^{2} (- 1 \cdot 1)}{9}$

Multiply $- 1$ by $1$ .

$- \frac{25 π^{2} \cdot - 1}{9}$

Multiply $- 1$ by $25$ .

$- \frac{- 25 π^{2}}{9}$

Move the negative in front of the fraction.

$- - \frac{(25) π^{2}}{9}$

Multiply $- - \frac{25 π^{2}}{9}$ .

Multiply $- 1$ by $- 1$ .

$1 \frac{25 π^{2}}{9}$

Multiply $\frac{25 π^{2}}{9}$ by $1$ .

$\frac{25 π^{2}}{9}$

Step 12

$x = - \frac{14}{5}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{14}{5}$ is a local minimum

Step 13

Find the y-value when $x = - \frac{14}{5}$ .

Replace the variable $x$ with $- \frac{14}{5}$ in the expression.

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{5 π}{6} \cdot ((- \frac{14}{5}) + 4))$

Simplify the result.

Simplify each term.

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{5 π}{6} \cdot (- \frac{14}{5} + 4 \cdot \frac{5}{5}))$

Combine $4$ and $\frac{5}{5}$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{5 π}{6} \cdot (- \frac{14}{5} + \frac{4 \cdot 5}{5}))$

Combine the numerators over the common denominator.

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{5 π}{6} \cdot \frac{- 14 + 4 \cdot 5}{5})$

Simplify the numerator.

Multiply $4$ by $5$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{5 π}{6} \cdot \frac{- 14 + 20}{5})$

Add $- 14$ and $20$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{5 π}{6} \cdot \frac{6}{5})$

Cancel the common factor of $5$ .

Factor $5$ out of $5 π$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{5 (π)}{6} \cdot \frac{6}{5})$

Cancel the common factor.

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{5 π}{6} \cdot \frac{6}{5})$

Rewrite the expression.

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{π}{6} \cdot 6)$

Cancel the common factor of $6$ .

Cancel the common factor.

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{π}{6} \cdot 6)$

Rewrite the expression.

$f (- \frac{14}{5}) = - 3 + 4 \cos (π)$

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.

$f (- \frac{14}{5}) = - 3 + 4 (- \cos (0))$

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

$f (- \frac{14}{5}) = - 3 + 4 (- 1 \cdot 1)$

Multiply $4 (- 1 \cdot 1)$ .

Multiply $- 1$ by $1$ .

$f (- \frac{14}{5}) = - 3 + 4 \cdot - 1$

Multiply $4$ by $- 1$ .

$f (- \frac{14}{5}) = - 3 - 4$

Subtract $4$ from $- 3$ .

$f (- \frac{14}{5}) = - 7$

The final answer is $- 7$ .

$y = - 7$

Step 14

These are the local extrema for $f (x) = - 3 + 4 \cos (\frac{5 π}{6} \cdot (x + 4))$ .

$(- 4, 1)$ is a local maxima

$(- \frac{14}{5}, - 7)$ is a local minima

Step 15

Find the Maximum/Minimum Value y=-3+4cos((5pi)/6(x+4))

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