Results 1 to 2 of 2

Math Help - Help with some proofs

  1. #1
    Newbie
    Joined
    May 2009
    Posts
    2

    Help with some proofs

    Suppose that f(x) has a continuous first derivative for all x R

    a) Prove that f(x) is concave if and only if f(x*)+(x-x*)f′(x*) ≥ f(x) for all x and x* R

    b)Given that f(x) is concave, prove that x* is a global maximum of f(x) is and only if f′(x*)= 0

    c) Given that f(x) is strictly concave, prove that it cannot possess more than one global maximum.

    my understanding of graphs is very poor any help on this would be greatly appreciated.

    Dave
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,413
    Thanks
    1852
    Quote Originally Posted by Rylude View Post
    Suppose that f(x) has a continuous first derivative for all x R

    a) Prove that f(x) is concave if and only if f(x*)+(x-x*)f′(x*) ≥ f(x) for all x and x* R
    Find the equation of the line from (x, f(x)) to (x*,f(x*)) and use the definition of concave.

    b)Given that f(x) is concave, prove that x* is a global maximum of f(x) is and only if f′(x*)= 0
    If f'(x)= 0, there are three possibilities: x* is a local maximum, a local minimum, or a point of inflection. Show that it must be a local maximum and then a global maximum.

    c) Given that f(x) is strictly concave, prove that it cannot possess more than one global maximum.
    Suppose it had two global maximum and show that leads to a contradiction. You might want to look at the line between the two maxima.

    my understanding of graphs is very poor any help on this would be greatly appreciated.

    Dave[/QUOTE]
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. proofs
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: March 2nd 2010, 04:54 AM
  2. lim sup and lim inf proofs
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: February 24th 2010, 08:02 PM
  3. More Proofs
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: February 13th 2008, 08:05 PM
  4. Proofs
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 3rd 2008, 05:23 AM
  5. Replies: 3
    Last Post: October 6th 2007, 03:01 PM

Search Tags


/mathhelpforum @mathhelpforum