Prove that a function is concave

[blaah]

Homework Statement

f:R->R, c'
prove that f is concave iff f(x*)+(x-x*)f'(x*)>=f(x)

Homework Equations

assume the function is only once differentiable

The Attempt at a Solution

i have no idea how to approach this question...

 

[Mark44]

Homework Statement

f:R->R, c'
prove that f is concave iff f(x*)+(x-x*)f'(x*)>=f(x)

Homework Equations

assume the function is only once differentiable

The Attempt at a Solution

i have no idea how to approach this question...

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

 

[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...

 

[HallsofIvy]

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