Second Derivative Test (2D)

Business Calculus · Axiom Academy

LESSON Second Derivative Test in 2D Classifying critical points using the discriminant To classify a critical point (a, b) of f(x, y), we compute the discriminant (also called the Hessian determinant): Evaluate all second partial derivatives at the critical point (a, b) The surface curves downward in all directions from this point. The surface curves upward in all directions from this point. The surface curves up in one direction and down in another. The test gives no information; other methods needed. Classify all critical points of Setting both to zero and solving: Since D (0, 0) is a saddle point Since D > 0 and f xx (1,1) = 6 > 0, (1, 1) is a local minimum f xx tells us: Is the surface curving up or down in the x-direction? D > 0 means: f xx and f yy have the same sign AND the twist (f xy ) isn't too strong D The curvatures in x and y directions disagree, creating a saddle D = 0 means: The situation is balanced on a knife's edge—could go either way Think of it like this: for a maximum or minimum, the surface must curve the same way in all directions. The discriminant D checks whether this consistency exists. After finding a critical point at (p*, q*) for a profit function: D > 0, f pp You found the profit-maximizing prices! ✓ D > 0, f pp > 0: This is a profit minimum (worst combination) D Saddle point—profit increases in one direction, decreases in another In practice, profit functions often have negative second partials (diminishing returns), leading to clean maxima.

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