Calculating Critical Points

To find all critical points of $f$, solve the system $\nabla f(\mathbf{x}) = \mathbf{0}$, i.e., all $n$ partial derivatives equal to zero simultaneously. This is an $n$-equation system in $n$ unknowns.

For polynomial or rational functions, this typically reduces to solving polynomial equations. The system may have zero, finitely many, or infinitely many solutions. Each solution is a critical point — but we must classify it further to determine if it is a local minimum, local maximum, or saddle point.

Example: for $f(x,y) = x^3 - 3x + y^2 - 2y$: $\partial f/\partial x = 3x^2 - 3 = 0 \Rightarrow x = \pm 1$ $\partial f/\partial y = 2y - 2 = 0 \Rightarrow y = 1$ Critical points: $(1, 1)$ and $(-1, 1)$.