Question #1869

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

Expert's answer

Take the derivatives in x and y:

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 = 0

y=0

x=0

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

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).

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