Maximum value of $x_1^2+x_2^2+\dots +x_k^2$

Question

While trying to solve a problem in graph theory I need to find the maximum value of $x_1^2+x_2^2+\dots +x_k^2$ subject to a condition that $x_1+x_2+\dots + x_k=n$.

Could someone guide me how to do this?

Thanks in advance.

Answer 1

There is no maximum. Consider for example the points $$(r,n-r,0,\ldots,0)$$ for $r\to\infty$.

However, there is a minimum, as pointed out in another answer.

Answer 2

The constraint describes a hyperplane perpendicular to the vector $(1,1,\cdots1)$. The isosurfaces of the objective function are hyperspheres centered at the origin.

Hence the solution

$$\frac nk(1,1,\cdots1).$$