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Question

Given f(x,y)=2x³-6x²-3y²+6xy-12x.

i) find critical points of f.

ii) use the second derivative test to determine whether each of the critical points is a local maximum, a local minimum or a saddle point.

Answer 1

f_x = 6x² - 12x + 6y - 12 = 0; f_y = - 6y + 6x = 0; x = y; x² - x - 2 = 0; x = -1, x = 2.

Two critical points : (-1, -1) and (2, 2)

f_xx = 12x - 12; f_yy = -6; f_xy = 6

(-1, -1): f_xx = -24; f_yy = -6; f_xy = 6; f_xx·f_yy - f_xy² = -24·(-6) - 36 = 108 > 0, max f(x,y) = f(-1, -1) = - 2 - 6 - 3 + 6 + 12 = 7.

(2, 2): f_xx = 12; f_yy = -6; f_xy = 6; ; f_xx·f_yy - f_xy² _{= - 72 - 36 = - 108 < 0; (2,2) is saddle point}