Results 1 to 2 of 2

Math Help - Mean Value Theorem Proof

  1. #1
    Junior Member
    Joined
    Mar 2009
    Posts
    41

    Mean Value Theorem Proof

    suppose that 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). [hint: Apply the Mean Value Theorem to the function h=f-g.]

    I know that h=f-g is continuous on [a,b] and differentiable on (a,b), but i dont know where to go from there. If you could help me it would be greatly appreciated. Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member Black's Avatar
    Joined
    Nov 2009
    Posts
    105
    If h = f-g, then it is continuous on [a,b] and differentiable on (a,b) (as you pointed out). So, by the MVT, there exists c in (a,b) s.t.

    h'(c)=\frac{h(b)-h(a)}{b-a}=\frac{h(b)}{b-a}.

    Since f'(x)<g'(x), h'(c) is negative. Therefore, h(b) must be negative, or f(b)<g(b).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. who can proof the lie's theorem?
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: December 28th 2011, 08:26 AM
  2. Proof of theorem...
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: January 31st 2010, 03:40 PM
  3. proof using mean value theorem
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: November 16th 2009, 05:50 PM
  4. Mean Value Theorem Proof
    Posted in the Calculus Forum
    Replies: 9
    Last Post: January 26th 2009, 08:15 PM
  5. proof of a theorem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 4th 2008, 07:36 PM

Search Tags


/mathhelpforum @mathhelpforum