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Homework Help: Using the definition of a concave function prove that it is concave (don't use dx/dy)

  1. Apr 22, 2012 #1
    Let D=[-2,2] and $$f:D\rightarrow R$$ be $$f(x)=4-x^2$$ Sketch this function.Using the definition of a concave function prove that it is concave (do not use derivative).

    Attempt:
    $$f(x)=4-x^2$$ is a down-facing parabola with origin at (0,4). I know that.

    Then, how do I prove that f(x) is concave using the definition of a concave function? I got the inequality which should hold for f(x) to be concave:

    For two distinct non-negative values of x u and v

    $$f(u)=4-u^2$$ and $$f(v)=4-v^2$$

    Condition for a concave function:

    $$ \lambda(4-u^2)+(1-\lambda)(4-v^2)\leq4-[(\lambda u+(1-\lambda)v]^2$$

    After expanding the inequality above I get:

    $$(\lambda u-\lambda v)^2\leq(\sqrt{\lambda} u-\sqrt{\lambda} v)^2$$

    I do not know what to do next.
     
    Last edited: Apr 22, 2012
  2. jcsd
  3. Apr 22, 2012 #2

    SammyS

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    Re: Using the definition of a concave function prove that it is concave (don't use dx

    You have to show that the above is true, not assume it's true.

    Isn't it also required that 0 ≤ λ ≤ 1 ?
     
  4. Apr 22, 2012 #3
    Re: Using the definition of a concave function prove that it is concave (don't use dx

    Yeah it is required that 0 ≤ λ ≤ 1.

    But, how do I prove that the below inequality is true?:

    $$λ(4−u)^2+(1−λ)(4−v)^2≤4−[(λu+(1−λ)v]^2$$
     
  5. Apr 22, 2012 #4

    SammyS

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    Re: Using the definition of a concave function prove that it is concave (don't use dx

    Without Loss of Generality, let u ≥ v .

    What else do you know about u & v ?
     
  6. Apr 22, 2012 #5
    Re: Using the definition of a concave function prove that it is concave (don't use dx

    That they are non-negative. I was working on this problem for a long time and I managed to turn the inequality:

    $$ \lambda(4-u^2)+(1-\lambda)(4-v^2)\leq4-[(\lambda u+(1-\lambda)v]^2$$

    into

    $$\lambda (u-v)^2(1-\lambda)\leq0$$

    Since 0<λ<1 the above inequality is true and the function f(x) is concave. I think it is correct.
     
  7. Apr 22, 2012 #6

    SammyS

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    Re: Using the definition of a concave function prove that it is concave (don't use dx

    Why do you say u & v are non-negative?
     
  8. Apr 23, 2012 #7
    Re: Using the definition of a concave function prove that it is concave (don't use dx

    Oh no they are not. My mistake. I was looking at another problem I am doing right now. It does not really matter whether they are negative or not for this inequality since the power of 2 makes their difference positive. But yeah they can be negative since they are distinct values of x and the the domain of x is [-2,2].
     
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