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Differantiation proof question

  1. Mar 11, 2009 #1
    f(x) is a differentiably continues on [-1,1]
    and differentiable twice on (-1,1)
    suppose f(0)=-1 and f(1)=1 and function f(x) has on (-1,0) an extreme point.

    prove that there is a point c on (-1,1) so f''(c)>=1
    ??

    prove:
    because of ferma law
    if a function has an extreme point on (-1,0) on the point c1 in (-1,0) then
    f(x) differentiable on (c1) and f'(c)=0

    because of mean value theorem there is a point c2 on (0,1):
    f'(c2)=[f(1)-f(0)]/(1-0)=2
    and because f'(x) differentiable once more we can do MVT again
    f''(c3)=[f'(c2)-f'(c1)]/c2-c1=(2-0)/(c2-c1)
    c1 and c2 are inside (-1,1) so the difference between them is smaller then 2
    f''(c3)=2/(c2-c1)>=2/2 >=1

    i cant understand why do i need to know that its differentiable continuously
    it seem like the fact that it differentiable twice is enough
    ??
     
  2. jcsd
  3. Mar 11, 2009 #2

    CompuChip

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    To apply the MVT to f' you need that f' is differentiable on (-1, 1) AND continuous on [-1, 1].
    This is precisely equivalent to continuously differentiable on [-1, 1] and twice differentiable on (-1, 1), as you can check.
     
  4. Mar 11, 2009 #3
    so we need f(x) to be continuously differentiable in order to do mvt for f'(x)

    but we did it twice
    we have not been given that f'(x) continuously differentiable too
    so we didnt have the right to do mvt on f''(x)
    ??
     
  5. Mar 11, 2009 #4

    CompuChip

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    Check again the requirements for the MVT.

    We need f to be continuous on the closed interval and differentiable on the open interval. There are no requirements on the continuity of the derivative. So if you want to apply the MVT to f', it is enough if f' is continuous on the closed interval and differentiable on the open interval, which means that f must be continuously differentiable on the closed interval and twice differentiable (without requirement on the continuity of the second derivative) on the open interval.
     
  6. Mar 11, 2009 #5
    thanks :)
     
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