Given for example a function of 2 variables $z = f(x,y) = x^2 + xy + y^2$, I believe there are infinitely many directions on the $xy$ plane in which we could take partial derivatives (this could already be wrong...).
If we differentiate $z$ w.r.t $x$, we get $f_x = 2x+y$. If we set this partial derivative equal to zero, we get infinitely many solutions; namely it seems that for any value of $y$, we could find a value of $x$ such that at the point $(x,y)$ the slope in the direction of the $x$ axis would be zero.
However, only one of these infinitely many points is actually the minimum of the function.
But if we take the other partial derivative $f_y = 2y + x$ and set both of them equal to zero as a system of equations and solve that system, we will get our minimum point at $(0,0)$.
Why is that? Why is it that we need both partial derivatives and not fewer, or especially more?
EDIT: Just to make it clear, I'm wondering both why two PD's are the minimum, and also why they are guaranteed to be enough (as in, why don't we need 20 PD's?).