Let $x_1+x_2+\cdots+x_n=m$ then minimize the function
$f(x_1,x_2,\cdots,x_n)=\sum_{i=1}^n (x_i)^{\alpha}$
where $x_i,m,n$ are positive integers and $\alpha>1$.
My attempt: I applied the Lagrange's multiplier and found that minimum is obtained when all $x_i's$ are equal, but I am unable to prove that these $x_i's$ are integers.
My observation: I think the minimum is obtained when all $x_i's$ are almost equal i.e. $|x_i-x_j|\le1, i,j=1,2,\cdots,n$.