Question

Given f(x,y)=xye[sup]-(x^2+y^2)/2[/sup],& classify its stationary points.

Answer 1

Take the derivatives in x and y:

f'_x = y* exp(-(x²+y²)/2) + x² * y * exp(-(x²+y²)/2) = (1+x²)* y * exp(-(x²+y²)/2)

similarly,

f'_y = (1+y²)* x * exp(-(x²+y²)/2)


The equation for stationary points:

f'_x = f'_y = 0.

Thus we have a system
(1+x²)* y * exp(-(x²+y²)/2)=0
(1+y²)* x * exp(-(x²+y²)/2)=0

(1+x²)* y = 0
(1+y²)* x = 0

y=0
x=0

Hence f(x,y) has a unique stationary point (0,0).