This is a continuation of the question asked in second-order divided differenc.
Suppose the periodic function $f(t)$ has only one value, $f(t_0) = -10$ within the period $[0,T]$, all other values of the function being zero within that period. We can interpolate $f(t)$ by $asin(t) - 10$ which for $t_0 = 0$ will give the value $-10$. Now, if we accept the interpolation then $f'(t) = acos(t)$ and therefore for $t_0=0$ we will have $f'(t_0) = acos(0) = a$. Thus, at $t_0 =0$ we will obtain $f(t_0)f'(t_0) = -10a$. At the end of the period the value of $f(t)f'(t)$ will be also $-10a$. Therefore, the average value of $f(t)f'(t)$ for the period is $\frac{2(-10a)}{2} = -10a$. Do you agree with this?