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?
A different approach...
Let be integrable function on , then we can easily prove the following:
i)If for all then
Proof:
For every partition of and for chosen points, we eill have:
.
.
ii) If for all then
The proof is identical to proof in i).
Now, by combining i) and ii) we will get the next result:
If is integrable on , and let be the supremum of on and is the infinum of on . Then:
(1)
Now, to your question:
By dividing (1) by , we will get:
Now, using MVT we conclude the existence of such that: