R twice continuously differentiable function proof

[irresistible]

r twice continuously differentiable function proof...

Homework Statement

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

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

(integral going from a to b)

Homework Equations

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

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?

 

[Dick]

Here's a hint. The derivative condition tells you that f(x) is concave down on the interval [a,b]. Draw a graph. Draw the secant line connecting x=a and x=b. That's below the curve, right? Now draw the tangent line at x=(a+b)/2. That's above the curve, right? Your two bounds are related to the integrals of those two linear functions. You'll need the MVT to prove the aboveness and belowness parts, if you don't already have such theorems.

 

[irresistible]

the inequality makes sense now that you explained,
I still have trouble proving it,

Where do I use the fact that f is TWICE differentiable?Does that make a difference?

 

[Dick]

You need twice differentiable to show some properties of your curve related to it being concave down. Like those I pointed out yesterday. You haven't done anything on this problem yet. I'd suggest you get started. Don't PM me about problems, ok?