critical point

A point (a,b) in the interior of f's domain is a critical point if \nabla f(a,b) = \mathbf{0} — that is, f_x(a,b)=0 and f_y(a,b)=0 simultaneously — or if either partial derivative fails to exist there. Both partials vanishing is the two-variable analogue of f'(x)=0; the second case (a corner, a cusp, a vertical tangent plane) is the analogue of f'(x) not existing at a sharp point. Being critical is necessary but not sufficient for a local extremum: every local max or min of a differentiable f occurs at a critical point, but not every critical point is an extremum.