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Q)Given $x_1+x_2+...+x_n=a$ where $a>0$,
find the extremum value of $f(x_1,x_2,...,x_n)=x_1^k+x_2^k+...+x_n^k$
Also, find the range of $k$ in which the extremum value of $f$ is a maximum value or a minimum value.

I tried to use lagrange multiplyer, but I'm totally stucked at finding S.O.C.(I used Hessian matrix) And I'm not sure even F.O.C.s that I've found are right or not... Please explain me how to solve it.

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  • $\begingroup$ Could you please define SOC and FOC? $\endgroup$ – John Dec 10 '13 at 18:32
  • $\begingroup$ The easiest way is to use Jensen's inequality, noting how the convexity changes for different values of k. Consider positive and negative values, greater and less than one. $\endgroup$ – Bill Kleinhans Dec 10 '13 at 21:47

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