In a Problem of least squares aproximation of a function $f:\mathbb R\longrightarrow\mathbb R$, in an interval $[a, b]$ by a polynomial of degree $n$
$$p(x)=c_0+c_1x+c_2x^2+\ldots+c_nx^n,\;\;\;\;\;\;c_n\neq 0,$$
we have to minimize the function $\psi$ defined by $$\psi(c_0, c_1,\ldots, c_n)=\int_a^b|f(x)-p(x)|^2dx. $$
It is customary to solve the problem as follows:
Solve the system: $$\frac{\partial\psi}{\partial c_i}=0,\;\;\;\;for\;i=0,\ldots, n,$$
then the solution of this system is the minimum. My question is:
Why the point where the partial derivatives are zero is the minimum and not the maximum or an inflection point?
