1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Question regarding Gateaux Derivative

  1. Apr 7, 2013 #1
    1. The problem statement, all variables and given/known data
    I am trying to solve the following problem:

    Let [itex]X[/itex] be space of continuous functions on [0,1] and let [itex]F:X\rightarrow\mathbb{R}[/itex] be defined by [itex]F(f)=\max\limits_{0\leq x\leq 1} f(x)[/itex] for any [itex]f\in X[/itex]. Show that the Gateaux Derivative does not exist if [itex]f[/itex] achieves a maximum at two different points [itex]x_1,x_2[/itex] in [0,1].

    2. Relevant equations
    The Gateaux Derivative for [itex]f,h\in X[/itex] is given by

    [itex]\lim\limits_{t\to0}\frac{1}{t}\left(F(f+th)-F(f)\right)[/itex]

    if the above limit exists for any increment [itex]h[/itex].

    3. The attempt at a solution
    Using the limit definition of the Gateaux Derivative, we see that

    [itex]\lim\limits_{t\to0}\frac{1}{t}\left(F(f+th)-F(f)\right)
    =\lim\limits_{t\to0}\frac{1}{t}\left(\max\limits_{x}(f+th)(x)-\max\limits_{x}f(x)\right)[/itex]
    =[itex]\lim\limits_{t\to0}\frac{1}{t}\left(\max\limits_{x}f(x)+\max_{x}th(x)-\max\limits_{x}f(x)\right)[/itex]
    =[itex]\lim\limits_{t\to0}t\frac{\max\limits_{x}h(x)}{t}[/itex]
    =[itex]\max\limits_{x}h(x)[/itex].

    This seems to work regardless of whether or not the function has a unique maximum, so that is the part I don't understand. Any help would be appreciated.
     
    Last edited: Apr 7, 2013
  2. jcsd
  3. Apr 7, 2013 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    While I'm still not clear on the whole problem, I'll tell you one thing that's wrong. max(f(x)+th(x)) is not generally equal to max(f(x))+max(th(x)). Try it with a simple example like f(x)=x and h(x)=(-x).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Question regarding Gateaux Derivative
  1. Gateaux derivative (Replies: 14)

Loading...