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Homework Help: Can someone explain the steps to me please

  1. Jun 30, 2006 #1
    the question is if f and g are continuous on [a,b] and differentiable on (a,b). Suppose also that f(a)=g(a) and f'(x)<g'(x) for a<x<b. Prove that
    f(b)<g(b). The hint is to apply the mean value theorem to the function
    h=f-g
    So assume that h=f-g. Since f and g are continuous on [a,b] and differentiable on (a,b) the mean value theorem applies by which there has to be a number such that h(b)= h(b)-h(a)=h'(c)(b-a). Since h'(c)<0 then h'(c)(b-a)<0, so
    f(b)-g(b)=h(b)<0 and hence f(b)<g(b).
    in this explanation why is h(b)=h(b)-h(a) and why is h'(c)<O. :confused:


    The folly of mistaking a paradox for a discovery, a metaphor for a proof, a torrent of verbiage for a torrent of capital truths, and ourselves for an oracle, is inborn in us--Paul Valerey
     
  2. jcsd
  3. Jun 30, 2006 #2

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    h(a)=f(a)-g(a)=0, and h'(c)<0 since h'(x)=f'(x)-g'(x)<0 for all x between a and b.
     
  4. Jun 30, 2006 #3
    Please bear with me. I am new to calculus. Will be starting college in the fall and am learning now on my own. Thank you for your help status!!!!!!
     
    Last edited: Jun 30, 2006
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