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Prove that a function is concave

  1. Oct 11, 2008 #1
    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...:confused:
  2. jcsd
  3. Oct 11, 2008 #2


    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.

  4. Oct 12, 2008 #3
    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...
  5. Oct 12, 2008 #4


    User Avatar
    Staff Emeritus
    Science Advisor

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