# 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'

similarly,

f'

The equation for stationary points:

f'

Thus we have a system

(1+x

(1+y

(1+x

(1+y

y=0

x=0

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

f'

_{x}= y* exp(-(x^{2}+y^{2})/2) + x^{2}* y * exp(-(x^{2}+y^{2})/2) = (1+x^{2})* y * exp(-(x^{2}+y^{2})/2)similarly,

f'

_{y}= (1+y^{2})* x * exp(-(x^{2}+y^{2})/2)The equation for stationary points:

f'

_{x}= f'_{y}= 0.Thus we have a system

(1+x

^{2})* y * exp(-(x^{2}+y^{2})/2)=0(1+y

^{2})* x * exp(-(x^{2}+y^{2})/2)=0(1+x

^{2})* y = 0(1+y

^{2})* x = 0y=0

x=0

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

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