Understanding Twice Differentiable Functions with f''(x) ≥ 0

[stukbv]

Homework Statement

1. if a function is twice continuously differentiable with f''(x) >= 0 for all real values of x then

(f(-x) + f(x))/2 >= f(0) ?


2. if a function is twice continuously differentiable with f''(x) >= 0 for all real values of x then
tf(x) + (1-t)f(y) >= f(tx+(1-t)y)
for all real values of x and y and for 0<=t<=1


I am really confused with these types of questions and to how to attempt them in my exam.

Thanks

 

[jbunniii]

If f&#039;&#039;(x) \geq 0 for all real x, then what kind of function is f?

 

[stukbv]

convex?

 

[jbunniii]

convex?

Yes. Now what's the definition of a convex function?

 

[stukbv]

No idea?

 

[micromass]

No idea?

Look it up?

 

[stukbv]

when i try and look it up it just keeps telling me exactly what part 2 says, i know its true but i need to prove it you see.

 

[jbunniii]

Yes, the conclusion of part 2 is exactly the definition of a convex function.

So part 2 is asking, true or false, f''(x) >= 0 for all x implies that f is convex. This is true, and the proof is a standard one which should be in your calculus book under "second derivative test" or something similar.

What about part 1?