# Second derivative positive implikes midpoint convex

1. Dec 21, 2011

### resolvent1

I've been trying to use Taylor's theorem with h = (y-x)/2 to show that a twice differentiable function for which the second derivative is positive is midpoint convex (ie, f( (1/2)*(x+y) ) \leq (1/2) * (f(x)+f(y)) ). (It's not a homework problem.) The problem I end up with this is that I'm not sure how to show that the sum of the terms involving the first and second derivative is nonpositive. How would I go about showing this, or is there a better (non-Taylor) way to do it?

2. Dec 22, 2011

### Office_Shredder

Staff Emeritus
Taylor series seems overwhelming for this kind of problem. I would try using a mean value theorem styled argument (post again if you need further guidance)

3. Dec 22, 2011

### resolvent1

THanks, I've got it.