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Answer to Question #1869 in Calculus for fergie

Question #1869
Given f(x,y)=xye-(x^2+y^2)/2,& classify its stationary points.
Expert's answer
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).

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