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R twice continuously differentiable function proof

  1. Feb 10, 2009 #1
    r twice continuously differentiable function proof....

    1. The problem statement, all variables and given/known data
    Help :frown:
    if f:[a,b] [tex]\rightarrow[/tex] R is twice continuously differentiable, and f(x)[tex]\geq[/tex] 0 for all x in [a.b]
    and f ''(x) [tex]\leq[/tex] 0 for all x in [a,b]
    prove that

    1/2 (f(a) + f(b)) (b-a) [tex]\leq[/tex] [tex]\int[/tex] f(x)dx [tex]\leq[/tex](b-a) f(a+b/2)

    (integral going from a to b)



    2. Relevant equations

    -It's given that f is twice continuously differentiable.




    3. The attempt at a solution
    By looking at the inequality I'm assuming that I have to use the mean value theorem somewhere..
    But how?
     
  2. jcsd
  3. Feb 10, 2009 #2

    Dick

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    Re: r twice continuously differentiable function proof....

    Here's a hint. The derivative condition tells you that f(x) is concave down on the interval [a,b]. Draw a graph. Draw the secant line connecting x=a and x=b. That's below the curve, right? Now draw the tangent line at x=(a+b)/2. That's above the curve, right? Your two bounds are related to the integrals of those two linear functions. You'll need the MVT to prove the aboveness and belowness parts, if you don't already have such theorems.
     
  4. Feb 11, 2009 #3
    Re: r twice continuously differentiable function proof....

    the inequality makes sense now that you explained,
    I still have trouble proving it,

    Where do I use the fact that f is TWICE differentiable?Does that make a difference?
     
  5. Feb 11, 2009 #4

    Dick

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    Re: r twice continuously differentiable function proof....

    You need twice differentiable to show some properties of your curve related to it being concave down. Like those I pointed out yesterday. You haven't done anything on this problem yet. I'd suggest you get started. Don't PM me about problems, ok?
     
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