Results 1 to 2 of 2

Math Help - prove: as the function increases then the graph is concave up/down

  1. #1
    Newbie
    Joined
    Oct 2008
    From
    NY
    Posts
    7

    prove: as the function increases then the graph is concave up/down



    how do I even start this one?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member bkarpuz's Avatar
    Joined
    Sep 2008
    From
    R
    Posts
    481
    Thanks
    2
    Quote Originally Posted by tehbrosta View Post


    how do I even start this one?
    I will just give the solution for 2. (a) since the remaining case is very similar.
    Clearly, if f is increasing then we know that f(x)\geq f(a) for all x\in[a,b].
    Hence, we have
    \int\limits_{a}^{b}f(x)dx\geq\int\limits_{a}^{b}f(  a)dx=f(a)(b-a),
    which completes the first part of the inequality.
    For the remaining part, we consider that f is concave down (convex).
    This indicates that f(x)\leq\frac{f(b)-f(a)}{b-a}(x-a)+f(a) holds for all x\in[a,b] (draw a graph for the increasing convex function f and the line passing at the points (a,f(a)) and (b,f(b))).
    Integrating both sides of this inequality, we get
    \int\limits_{a}^{b}f(x)dx\leq\int\limits_{a}^{b}\B  igg(\frac{f(b)-f(a)}{b-a}(x-a)+f(a)\Bigg)dx=(b-a)\frac{f(b)+f(a)}{2},
    which completes the proof.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: July 27th 2011, 09:55 PM
  2. Entropy increases as number of bins increases
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: February 24th 2011, 12:14 PM
  3. Replies: 1
    Last Post: August 24th 2010, 07:06 AM
  4. Prove x^1/2 is concave
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 16th 2010, 09:08 AM
  5. graph that increases or decreases
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: July 9th 2006, 12:33 AM

Search Tags


/mathhelpforum @mathhelpforum