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Intro to analysis proof first and second derivatives and mean value theorem

  1. Feb 12, 2010 #1
    1. The problem statement, all variables and given/known data

    Let f(x) be a twice differentiable function on an interval I. Let f''(x)>0 for all x in I and let f'(c)=0 for some c in I. Prove f(x) is greater than or equal to f(c) for all x in I.

    2. Relevant equations

    Mean value theorem?

    3. The attempt at a solution

    f''(x)>0 implies that f'(x) is strictly increasing on I. I don't know what to use the f'(c)=0 for, and whether or not to use the mean value theorem.
     
  2. jcsd
  3. Feb 12, 2010 #2
    What does f''(x) indicate about f(x)? When you see this, you will realize how the f'(c)=0 comes in to play.
     
  4. Feb 12, 2010 #3
    It implies that f(x) is concave up?

    It also implies c is a minimum by the second derivative test but we haven't covered that as of this section in our textbook.
     
  5. Feb 13, 2010 #4
    I still can't figure this one out - I get the feeling the proof is much more simple than I think it is.
     
  6. Feb 13, 2010 #5

    Dick

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    Apply the MVT. If f(c) is not a minimum then there is a d such that f(d)<f(c). Suppose d<c. What does the MVT tell you? You are right, the proof is not that hard.
     
  7. Feb 13, 2010 #6
    The MVT tells me that there exists some Xo in [d,c] such that f'(Xo) = [f(c)-f(d)]/(c-d).
     
  8. Feb 13, 2010 #7

    Dick

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    Sure. Is that quantity positive or negative?
     
  9. Feb 13, 2010 #8
    It is negative. I'm still a bit puzzled as to how the mean value theorem relates to proving this problem.
     
    Last edited: Feb 13, 2010
  10. Feb 13, 2010 #9

    Dick

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    I don't agree with that. We assumed f(d)<f(c) and d<c. I think it's positive. You'll see how this problem relates to the MVT shortly. Feel free to jump ahead if you like.
     
    Last edited: Feb 13, 2010
  11. Feb 13, 2010 #10
    Ah my mistake, the quantity is positive. I think I know where this is headed.

    So f'(xo)>f'(c)=0 and c>xo. But f'(x) must be strictly increasing on I since f''(x)>0, and thus the statement must be true by contradiction.
     
  12. Feb 13, 2010 #11

    Dick

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    Yes, exactly. That's where the f'(c)=0 comes in. And if d>c?
     
  13. Feb 13, 2010 #12
    If d>c then f'(xo)<0 and xo>c. f'(c)=0 and f'(xo)<0. But this contradicts that f'(x) is strictly increasing. So the statement must be true.
     
  14. Feb 13, 2010 #13

    Dick

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    So you have it then.
     
  15. Feb 13, 2010 #14
    Thank you so much for your help!
     
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