# Real Analysis proof continuity

1. Nov 10, 2007

### CrazyCalcGirl

1. The problem statement, all variables and given/known data
Suppose that the function f is continuous on [a,b] and X1 and X2 are in [a,b]. Let K1 and K2 be positive real numbers. Prove that there exist c between X1 and X2 for which

f(c) = (K1f(X1) + K2f(X2))/(K1+k2)

2. Relevant equations

3. The attempt at a solution I know I am supposed to have attempted this before asking for help, but honestly I have looked at this problem for over an hour and cannot figure out what to use. I have tried looking at it in terms of Rolles Thrm, Intermediate Value Thrm, and Mean Value Thrm, but nothing is clicking. I also tried manipulating it just by moving things around. I don't seem to be getting anywhere and I'm just hoping someone can point me in the right direction. Then at least I can come up with a decent attempt.

2. Nov 10, 2007

### quasar987

Have you noticed that according the the intermediate value thm, all you need to do is to prove that

f(X1) $\leq$ (K1f(X1) + K2f(X2))/(K1+K2) $\leq$ f(X2) ?

Supposing f(X1) $\leq$ f(X2).

3. Nov 11, 2007

### CrazyCalcGirl

oh god. I feel dumb now. I'm not even going to post back. I got it.
Thank you.