Results 1 to 4 of 4

Math Help - Help with the interval question

  1. #1
    Member
    Joined
    Oct 2009
    From
    Canada
    Posts
    128

    Help with the interval question

    Assum that f and g are differentiable on the interval (-c, c) and f(0) = g(0).

    show that if f'(x) > g'(x) for all x \in (0,c), then f(x) > g(x) for all x \in(0,c).
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    By the properties of the integral we have \int_{0} ^{x} f'(t)dt > \int_{0}^{x} g'(t)dt which by the FTC is equivalent to f(x)-f(0)>g(x)-g(0) and since f(0)=g(0) we have f(x)>g(x) for all x \in (0,c)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2009
    From
    Canada
    Posts
    128
    Quote Originally Posted by Jose27 View Post
    By the properties of the integral we have \int_{0} ^{x} f'(t)dt > \int_{0}^{x} g'(t)dt which by the FTC is equivalent to f(x)-f(0)>g(x)-g(0) and since f(0)=g(0) we have f(x)>g(x) for all x \in (0,c)
    Sorry, We havent cover the integral material yet, we need to prove it by the mean value theorem or increasing or decreasing function properties.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Assume there is d\in (0,c) such that (f-g)(d)=0 then by Rolle's theorem there exists a point x_0\in (0,d) such that (f-g)'(x_0)=0 which is a contradiction since (f-g)'(x)>0 in (0,c) and this also implies that in a right neighbourhood of 0 we have that f-g is positive. And this is enough because since f-g is continous, if it were negative there would exist a point in which it's zero and we repeat the argument. So f-g is positive in (0,c)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Question on interval of convergence
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 24th 2011, 07:21 AM
  2. Confidence Interval Question
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: May 8th 2010, 11:53 PM
  3. ODE Question with t-interval
    Posted in the Calculus Forum
    Replies: 0
    Last Post: November 2nd 2009, 01:37 PM
  4. Interval Question
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 11th 2009, 08:01 AM
  5. Interval question..
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 23rd 2008, 06:48 PM

Search Tags


/mathhelpforum @mathhelpforum