Finding Extrema of f(x,y) = x³ - 3xy + y³

Calculus 3 · Axiom Academy

EXAMPLE Finding Extrema of f(x,y) = x^3 - 3xy + y^3 Locate every critical point, then classify it with the second-derivative (Hessian) test Find all critical points of f(x,y) = x^3 - 3xy + y^3 and classify each one as a local maximum, a local minimum, or a saddle point. The nested green loops close around the local minimum at (1,1) ; the level curve through the origin crosses itself, the signature of the saddle at (0,0) . The dashed line is y=x , which passes through both critical points. Nice work — you found and classified every critical point. Here is the method to carry forward: Critical points: set both first partials to zero, f_x = f_y = 0 , and solve the system simultaneously. The discriminant: measures how the surface curves at a critical point. Second-derivative test: and → local minimum Our results: (0,0) is a saddle point and (1,1) is a local minimum with f(1,1) = -1 . This routine works for any smooth function of two variables. Solving the system accurately is where most of the errors hide, so check that algebra carefully.

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