Sec. 5.2
#6a) grad(f) = (6x-2y,-2x+2y). Critical point at (0,0).
D = f_xx*f_yy - (f_xy)^2 = 12-4 = 8 and f_xx >0, so (0,0) is local minimum.
#6g) grad(f) = (-2x/(x^2+y^2_1)^2, -2y/(x^2+y^2+1)^2). Critical point
at (0,0). D = 4-0, and f_xx <0, so (0,0) is a local minimum.
Sec. 5.3
#3a) There are four solutions: first: x = 0, y = 1, lambda = -2;
second: x = 0, y = -1 lambda = -2;
third: x= -1, y = 0, lambda = 1; and fourth: x = 1, y = 0, lambda = 1.
#3b) There are four solutions: first: x= 0, y = 1, lambda = 1/2; second:
x = 0, y = 01, lambda = 1/2; third: x = sqrt(2), y = 0, lambda = 1; and fourth:
x = -sqrt(2), y = 0, lambda = 1.
#7a) Two solutions: first: lambda = 1, y=0 and x^2+z^2 = 1; second:
lambda = -1, x = 0 , z = 0, and y = +1 or -1.
#7b) Two solutions: first: lambda = -1, x = 0, y^2+z^2 = 1; second:
lambda = 1/3, y = 0, z = 0, x = sqrt(3) or -sqrt(3).
Last modified: Mon Nov 4 16:44:48 1996