In Problem use Theorem 2 to find the local extrema.

f(x, y) = x³ + y³ - 3xy

Transcribed Image Text:

THEOREM 2 Second-Derivative Test for Local Extrema If 1. z = f(x, y) 2. fx(a, b) = 0 and f(a, b) = 0 [(a, b) is a critical point] 3. All second-order partial derivatives of fexist in some circular region containing (a, b) as center. 4. A = f(a, b), B = fxy(a, b), C = fyy(a, b) Then Case 1. If AC - - Case 2. If AC - B² > 0 and A < 0, then f(a, b) is a local maximum. B² 0 and A > 0, then f(a, b) is a local minimum. B² < 0, then fhas a saddle point at (a, b). Case 3. If AC = Case 4. If AC - B² = 0, the test fails.