I have been asked to prove the following:

Suppose that f is continuous on [a,b]. Prove that there exists a point c such that the integral from a to be is f(c)(b-a).

Proof:

Let F(x)= integral from a to x of f(t)dt, then by the Mean Value Theorem there exists a value f(b) such that a b x. Furthermore, the integral from a to b of f(t)dt can be calculated to be f(c)(b-a).

This is all I have so far, I realize there are holes. Can you please help me with this?