Finding critical points of a multivariable function

In the same way we have to rely on derivatives to find critical points of a single variable functions, we have to rely on partial derivatives to find critical points of a multivariable function. The idea of looking for points were we have horizontal tangent lines or zeroes of a function remains the same for multivariable functions. Except that for two variables we have a plane, not just a single line.

Let [math]\displaystyle{ f }[/math] be a function with a domain [math]\displaystyle{ D }[/math]. [math]\displaystyle{ P_0 \in D }[/math] is a point of maximum or minimum. If [math]\displaystyle{ f }[/math] is differentiable at [math]\displaystyle{ P_0 }[/math], then its first order partial derivatives are equal to zero at that point.

Suppose that [math]\displaystyle{ (x_0,y_0) }[/math] is a local maximum of [math]\displaystyle{ f }[/math]. With [math]\displaystyle{ (x_0,y_0) }[/math] being inside [math]\displaystyle{ D }[/math], excluding the boundary, there exists an open ball [math]\displaystyle{ B \subset D }[/math], centered at [math]\displaystyle{ (x_0,y_0) }[/math], such that, for all [math]\displaystyle{ (x,y) }[/math] in [math]\displaystyle{ B }[/math]:

[math]\displaystyle{ f(x,y) \leq f(x_0,y_0) }[/math]