then at what value of x in the interval is f(x) a maximum?

By the Fundamental Theorem of Calculus,

Setting f’(x) = 0 yields the following three critical values.

Testing the values of f’(x) on the intervals [0,Pi] and [Pi,2Pi], shows that f’(x) goes from a positive value to a negative value at Pi, which implies that f(x) has a maximum value at x = Pi in the closed interval [0,2Pi]