How do I apply the Mean Value Theorem to this problem?

Question

f(x) = 2sinx + sin2x, [0,π]

(hint: cos2x = 2cos²x-1)

Answer 1

You're asked to find a value for x in the interval (0,π) such that

f'(x) = [f(π) - f(0)]/(π-0) = 0

Since the given f(x) is differentiable on [0,π], the theorem guarantees that at least one value exists (it doesn't help you find it).

f'(x) = 2 cos(x) + 2 cos(2x)

Find a value of x for which f'(x) = 0

To do this, use the hint; you'll get a quadratic to solve for cos(x).There are two solutions; one of them will not be of use, because it will not produce a value for x in (0,π).

You should get x = π/3