Given the function f(x,y) = e^(x^2 + y^2 + 2x - 2y), Classify each as being a minimum, maximum, or neither using D-test

Answer 1

f(x,y) = e^{(x^2+y^2+2x-2y)} > 0 for all points (x,y) in lR²

First, find the critical points:

f_x = (2x+2)e^{(x^2+y^2+2x-2y)} = 0 when x = -1

f_y = (2y-2)e^{(x^2+y^2+2x-2y)} = 0 when y = 1

The only critical point is (-1,1)

Since f_xx(x,y) = 2e^{(x^2+y^2+2x-2y)} + (2x+2)²e^{(x^2+y^2+2x-2y)}, f_xx(-1,1)) = 2e^-2 > 0

Since f_yy(x,y) = 2e^{(x^2+y^2+2x-2y)} + (2y-2)²e^{(x^2+y^2+2x-2y)} , f_yy(-1,1) = 2e^-2

Since f_xy(x,y) = (2x+2)(2y-2)e^{(x^2+y^2+2x-2y)}, f_xy(-1,1) = 0

D = f_xx(-1,1)f_yy(-1,1) - [f_xy(-1,1)]² = 4e^-4 > 0

Since D > 0 and f_xx(-1,1) > 0, there is a relative minimum at (-1,1).