Write as a function.

The derivative of with respect to is .

Since is constant with respect to , the derivative of with respect to is .

The derivative of with respect to is .

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

Multiply each term in by .

Multiply .

Multiply by .

Multiply by .

Multiply by .

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

The exact value of is .

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.

Subtract from .

The solution to the equation .

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

The exact value of is .

Multiply by .

is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

is a local maximum

Replace the variable with in the expression.

Simplify the result.

The exact value of is .

The final answer is .

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

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.

The exact value of is .

Multiply by .

Multiply by .

is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

is a local minimum

Replace the variable with in the expression.

Simplify the result.

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.

The exact value of is .

Multiply by .

The final answer is .

These are the local extrema for .

is a local maxima

is a local minima

Find the Local Maxima and Minima y=cos(x)