(adsbygoogle = window.adsbygoogle || []).push({}); r twice continuously differentiable function proof....

1. The problem statement, all variables and given/known data

Help

if f:[a,b] [tex]\rightarrow[/tex] R is twice continuously differentiable, and f(x)[tex]\geq[/tex] 0 for all x in [a.b]

and f ''(x) [tex]\leq[/tex] 0 for all x in [a,b]

prove that

1/2 (f(a) + f(b)) (b-a) [tex]\leq[/tex] [tex]\int[/tex] f(x)dx [tex]\leq[/tex](b-a) f(a+b/2)

(integral going from a to b)

2. Relevant equations

-It's given that f is twice continuously differentiable.

3. The attempt at a solution

By looking at the inequality I'm assuming that I have to use the mean value theorem somewhere..

But how?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# R twice continuously differentiable function proof

**Physics Forums | Science Articles, Homework Help, Discussion**