# Prove that a function is concave

1. Oct 11, 2008

### blaah

1. The problem statement, all variables and given/known data
f:R->R, c'
prove that f is concave iff f(x*)+(x-x*)f'(x*)>=f(x)

2. Relevant equations
assume the function is only once differentiable

3. The attempt at a solution
i have no idea how to approach this question...

2. Oct 11, 2008

### Staff: Mentor

Are x* and x any two values of x? Are there any restrictions on the values of x?

To prove your statement you need to prove two things:
1. f is concave ==> f(x*) + (x - x*) f'(x*) >= f(x)
2. f(x*) + (x - x*) f'(x*) >= f(x) ==> f is concave
For the first, what does it mean for a function to be concave?
For the second, one approach would be a proof by contradiction. Suppose that f(x*) + (x - x*) f'(x*) >= f(x) is true and that f is not concave. If you arrive at a contradiction, it means that your original assumption was incorrect, and therefore f must be concave.

Mark

3. Oct 12, 2008

### blaah

for all x, x*

i know that for the function to be concave all the points on the tangent need to be on or below the function....but i doesn't help....i've been staring at the problems for days now, with no result...

4. Oct 12, 2008

### HallsofIvy

Staff Emeritus
Looks to me like the mean value theorem would be useful here.