Critical Points of Multivariable Functions

Core Idea

A critical point of f(x, y) is a point where ∇f = 0 (both partial derivatives are zero) or where ∇f is undefined. Critical points are candidates for local maxima, local minima, and saddle points. Unlike single-variable calculus, critical points in ℝ² can be saddle points — points that are local minima in one direction and local maxima in another, with no extreme value. Finding critical points requires solving a system of equations f_x = 0 and f_y = 0 simultaneously.

How It's Best Learned

The saddle point concept has no single-variable analogue and requires geometric visualization. Show the surface z = x² − y² (a classic saddle) and identify that its critical point at the origin is neither a max nor a min. Then contrast with z = x² + y² (paraboloid) whose critical point at the origin is a minimum.

Common Misconceptions

• Not every critical point is a local extremum — saddle points are critical points that are neither maxima nor minima.
• Setting f_x = 0 alone is insufficient; both partial derivatives must be zero (or undefined) simultaneously.
• A function defined on a closed bounded domain can also attain its extrema on the boundary, not just at interior critical points.

Explainer

In single-variable calculus, you found candidates for local extrema by solving f′(x) = 0. The gradient ∇f you have studied is the multivariable generalization of f′: a vector whose components are the partial derivatives in each coordinate direction. A critical point of f(x, y) is a point where ∇f = 0 — that is, where both f_x = 0 and f_y = 0 simultaneously. Just as f′(x) = 0 was necessary (but not sufficient) for a local extremum in one variable, ∇f = 0 is necessary (but not sufficient) in two or more variables.

The geometric reason is the same as in one dimension. The gradient points in the direction of steepest ascent; if ∇f ≠ 0 at a point, you can move in the direction of ∇f to increase f, or opposite to it to decrease f. So any point with a nonzero gradient cannot be a local max or min — you can always improve the function value from there. Only when ∇f = 0 — all directional derivatives vanish — is the point a true standstill and a candidate for an extremum.

But multivariable calculus introduces a qualitatively new phenomenon with no single-variable analogue: the saddle point. Consider f(x, y) = x² − y². At the origin, f_x = 2x = 0 and f_y = −2y = 0, so the origin is a critical point. Moving along the x-axis (setting y = 0), the function equals x², which increases away from the origin — the origin looks like a minimum in this direction. Moving along the y-axis (setting x = 0), the function equals −y², which decreases — the origin looks like a maximum in this direction. No neighborhood of the origin contains it as either a local max or a local min; it is a saddle. In single-variable calculus, a critical point that is not a max or min is an inflection point — a single, degenerate case. In two dimensions, saddle points are a robust, generic phenomenon.

Finding critical points in practice requires solving the system f_x = 0 and f_y = 0 simultaneously, often yielding multiple solutions. Once you have the candidates, the second derivatives test (which builds on this topic) tells you which are maxima, which are minima, and which are saddles. For functions on a closed bounded domain, remember that global extrema may occur on the boundary rather than at interior critical points — boundary analysis must be combined with interior critical point analysis to find the true global max and min.