Extreme values of a function

Question

$x_1,x_2,...,x_m\in R^n$ are given. How to find $u\in R^n$ such that $\sum^{m}_{i=1} (d(u,x_i))^2$ is minimal.

I tried this way:
If
$J(u)=\sum ^{m}_{i=1} (d(u,x_i))^2$
$x_i=(x^{1}_{i},x^{2}_{i},...,x^{n}_{i}), i=1,2,...,m$
$u=(u_1,u_2,...,u_n)$
$(d(u,x_i))^2=\sum^{n}_{k=1} (u_k-x^{k}_{i})^2, i=1,..., m$
I tried to find partial derivatives of $J$ and I got that
$\dfrac{\partial J}{\partial u_i}=2(mu_i-(x^{i}_{1}+...+x^{i}_{m})), i=1,...,n$

If this is ok, I should determine when these partial derivatives are equal to $0$ and such values for $x_i$ would be the coordinates of vector $J'(u)$. Then I should calculate matrix $J''(u)$ and check if it is positive definite. Is this a good way to solution?
Are partial derivatives correct?

So does that mean the minimizer is the center of gravity? – Maesumi Jun 04 '13 at 15:07 — Maesumi, Jun 04 '13 at 15:07

score 1 · Accepted Answer · answered Jun 04 '13 at 14:15

This all seems fine. A few things you may want to consider as well (just out of interest, they shouldn't really impact your working) is how changing just one coordinate of $u$ affects $J(u)$ as well as if you can interpret in a simple way where $u$ is in relation to all the $x_i$

Extreme values of a function

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