Maximum value of a function

  • Thread starter gummz
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  • #1
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Homework Statement



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).

Homework Equations



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

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.
 

Answers and Replies

  • #2
Ray Vickson
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Homework Statement



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).

Homework Equations



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

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.
OK, so what are these facts telling you?
 
  • #3
RUber
Homework Helper
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f(x) = 1/(1+|x|) + 1/(1+|x-a|), a>0
I get the value x = a/2, which yields f(a/2) = 0, which is obviously a minimum and not a maximum.
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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