Analyzing f(x,y) = x² - y²

Calculus 3 · Axiom Academy

EXAMPLE Classifying the critical point of f(x,y)=x^2-y^2 Use the gradient and the second-derivative (discriminant) test to show the origin is a saddle point. Consider the surface f(x,y) = x^2 - y^2 . Find its critical point, then use the second-derivative test to classify it. Confirm the geometry by describing the level curves x^2 - y^2 = k . Level curves of f(x,y)=x^2-y^2 Every contour is a hyperbola. For k>0 they open left–right (blue); for k<0 they open up–down (red). At k=0 the "hyperbola" degenerates into the crossed lines , which pass through the saddle point at the origin. Nice work. You located the critical point of f(x,y)=x^2-y^2 and classified it end to end with the second-derivative test. Critical point: forces , so the only critical point is the origin (0,0) . Second-derivative test: with , the discriminant is . Classification: D<0 means the origin is a saddle point — neither a local max nor a local min. Level curves: x^2 - y^2 = k are hyperbolas: opening along the x -axis for k>0 , along the y -axis for k<0 , and degenerating to the crossed lines at k=0 . The mismatched signs of f_ xx and f_ yy are the whole story: the surface rises in the x -direction and falls in the y -direction, which is exactly what a saddle does.

This is the written version of the interactive lesson above. See the full Calculus 3 course.