Critical Points and Classification

Calculus 3 · Axiom Academy

LESSON Critical Points and Classification Locate every peak, valley, and pass of a surface z = f(x,y) — set the gradient to zero, then let the second-derivative test sort them out. 1. Finding the Critical Points A critical point of f(x,y) is a point where the gradient vanishes, so the tangent plane there is horizontal. Since points in the direction of steepest ascent, marks a spot that is momentarily flat in every direction. 2. The Hessian and the Discriminant To classify a critical point we need curvature , and curvature lives in the second derivatives. Collect all four of them into the Hessian matrix H . Because the mixed partials agree, f_ xy = f_ yx , so H is symmetric. Every second partial derivative, at the critical point The discriminant is the determinant of the Hessian With the discriminant in hand, the classification is a quick lookup on the signs of D and f_ xx . Watch a bowl flatten and buckle into a saddle as D falls through zero. 4. Three Shapes, Three Verdicts Here are the three outcomes on their canonical surfaces, each with its one critical point at the origin. Watch the two principal cross-sections — the slice along x and the slice along y — because their curvatures are exactly what D and f_ xx measure. You can now find a function's critical points and classify every one of them as a minimum, a maximum, or a saddle.

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