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Maximum value of a function

  1. Oct 6, 2014 #1
    1. The problem statement, all variables and given/known data

    f(x) = 1/(1+|x|) + 1/(1+|x-a|), a>0

    I am to show that the maximum value of this function is (2+a)/(1+a).

    2. Relevant equations

    None in particular. Derivatives for 1/x and the chain rule for f(g(x)).

    3. The attempt at a solution

    I have parted this function according to 0 <= x, 0 <= x <= a, a <= x, and differentiated and confirmed via Wolfram Alpha.

    The first derivative has no root for 0 <= x and a <= x, and for 0 <= x <= a I get the value x = a/2, which yields f(a/2) = 0, which is obviously a minimum and not a maximum.
     
  2. jcsd
  3. Oct 6, 2014 #2

    Ray Vickson

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    OK, so what are these facts telling you?
     
  4. Oct 6, 2014 #3

    RUber

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    Double check your evaluation at x=a/2. It seems like you took the right approach. Also, don't hesitate to plot the function with an arbitrary value a.
     
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