Define to be the average of a function on the interval ,
- If with , prove there exists with such that
- Prove part (a) holds for weighted averages of functions where for a nonnegative weight function we define the weighted average of on by
- Proof. Let
Then, since , we have . Furthermore,
So,
- Proof. Let
Then,
Thus, (since since and since is nonnegative both of these are nonnegative).
Then,
how did you find t? with such properties?
Just start with the average over the full range, then split it into (a, c) and (b, c). Then multiply the first part with (c-a)/(c-a) and the second one with (b-c)/(b-c). From there it should be easy to see.
It is similar for the weighted average, with a difference that the weight integral is positive (not only non-negative).