Fourier series minimize integral

Hi,

For this question the minimal values of a_i are just the coefficients of the Fourier series of f(x) but the comments mention that you need to differentiate with respect to a_i to find this, but I am not sure what to do there?

How might I differentiate this to find the minimal values of ai?

For example, for a0 youd find where
dF/da0 = 0
So differentiating under the integral youd have
2 (-1/2) Int [ f(x) - a0/2 + Sum ...] = 0
For each of the terms in the Sum the integral over the period is zero (harmonics) so
a0 = 2 Int f(x)

For the others its a similar argument.