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.