Prove that a function is concave

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  • #1
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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...:confused:
 

Answers and Replies

  • #2
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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...:confused:
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
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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
HallsofIvy
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Looks to me like the mean value theorem would be useful here.
 

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