Let f:[a,b]---->R be Riemann integrable. Define F:[a,b]---->R

by

F(x)=

iii) Prove that there exists c in[a,b] such that F(c)=sup{F(x): x in [a,b]}

By boundedness principle a continuous functions attains a maximum and minimum so f is bounded.

iv) Now suppose that f is continuous and that the point c from (iii) satisfies c in (a,b) what can you conclude about f(c)?

I know from lecture note,

If f is continuous at c in[a, b] then F is differentiable at c and F'(c) = f(c). from iii) f(c)=maximum of function so the derivative at maximum value is f'(c)=0 I have been asked for f(c) so can you help please? thanks