Local minimum of the function We have a function
$$\sum_{i=1}^{n}x_{i}^{2}=min,\quad\mbox{ subject to }\sum_{i=1}^{n}x_{i}=c,$$
which should have the minimizer
$$\frac{c^{2}}{n}.$$
However, from the Lagrangian
$$\Lambda\left(x_{i},\lambda\right)=\sum_{i=1}^{n}x_{i}^{2}+\lambda\left(\sum_{i=1}^{n}x_{i}-c\right),$$
is
$$x_{i}=-\frac{\lambda}{2}$$
which gives me the different minimizer
$$\frac{c}{n}.$$
Thanks for your help.

There is no conflict. The minimum value is $\frac{c^2}{n}$, attained at $x_i=\frac{c}{n}$. 
