,

By the Sum Rule, the derivative of with respect to is .

Evaluate .

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

The derivative of with respect to is .

Multiply by .

Evaluate .

Differentiate using the chain rule, which states that is where and .

To apply the Chain Rule, set as .

The derivative of with respect to is .

Replace all occurrences of with .

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

Differentiate using the Power Rule which states that is where .

Multiply by .

Move to the left of .

Set the first derivative equal to zero.

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor.

Factor by grouping.

Reorder terms.

For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .

Factor out of .

Rewrite as plus

Apply the distributive property.

Multiply by .

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

Factor the polynomial by factoring out the greatest common factor, .

Remove unnecessary parentheses.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

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

Set the first factor equal to and solve.

Set the first factor equal to .

Add to both sides of the equation.

Multiply each term in by

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 negative in the third and fourth quadrants. To find the second solution, subtract the solution from , to find a reference angle. Next, add this reference angle to to find the solution in the third quadrant.

Simplify the expression to find the second solution.

Simplify .

To write as a fraction with a common denominator, multiply by .

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Add and .

To write as a fraction with a common denominator, multiply by .

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Move to the left of .

Add and .

Subtract from .

The resulting angle of is positive, less than , and coterminal with .

The solution to the equation .

Set the next factor equal to and solve.

Set the next factor equal to .

Add to both sides of the equation.

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide 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.

Simplify .

To write as a fraction with a common denominator, multiply by .

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Move to the left of .

Subtract from .

The solution to the equation .

The final solution is all the values that make true.

Use the endpoints and all critical points on the interval to test for any absolute extrema over the given interval.

Evaluate the function at .

Simplify each term.

The exact value of is .

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Cancel the common factor of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

The exact value of is .

To write as a fraction with a common denominator, multiply by .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Move to the left of .

Add and .

Evaluate the function at .

Simplify each term.

The exact value of is .

Multiply by .

Multiply by .

The exact value of is .

Add and .

Evaluate the function at .

Simplify each term.

The exact value of is .

Multiply by .

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

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

The exact value of is .

Add and .

Compare the values found for each value of in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest value and the minimum will occur at the lowest value.

Absolute Maximum:

Absolute Minimum:

Find the Absolute Max and Min over the Interval f(t)=2cos(t)+sin(2t) , [0,pi/2]