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I have the following function

$f(x_1,x_2) = \frac{x_1}{x_2+p} + \frac{x_2}{x_1+p}$

where $x_1$ and $x_2$ $\in$ $[0,1]$ and $p > 0$ is a constant

I want to find global/local maxima for this. Please suggest some good methods. I already looked at stochastic ones (like PSO etc) but I am looking for other ways in which I can simplify my function.

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  • $\begingroup$ can $p > 0$ ?....... $\endgroup$
    – DeepSea
    Jul 28 '14 at 6:42
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    $\begingroup$ A necessary condition for f assumes a maximum or local minimum is $\frac{\partial f}{\partial x_1} = \frac{\partial f}{\partial x_2} = 0$. $\endgroup$
    – Mathsource
    Jul 28 '14 at 6:45
  • $\begingroup$ @8pir yes, i updated the question $\endgroup$
    – BaluRaman
    Jul 28 '14 at 6:46
  • $\begingroup$ Use the second partial test $\endgroup$ Jul 28 '14 at 12:16
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Hint: The function is convex in each of the variables, hence the extreme values occur at the end points of the interval. I hope that helps.

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  • $\begingroup$ How can you say it is convex in each variable? Can you please give some mathematical proof? $\endgroup$
    – BaluRaman
    Jul 28 '14 at 6:59
  • $\begingroup$ $\dfrac{\partial f^2}{\partial x_i} = \dfrac{2x_j}{(x_i+p)^3} > 0$ $\endgroup$
    – DeepSea
    Jul 28 '14 at 10:03

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