What is the extreme f(x,y)=e^{-y}(x^2+y^2)+6 ?

The extreme f(x,y)=e^{-y}(x^2+y^2)+6 is Minimum(0,0),Saddle(0,2)

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extreme f(x,y)=e^{-y}(x^2+y^2)+6

extreme points

f (x, y) = e^{- y} (x^{2} + y^{2}) + 6

Minimum (0, 0), Saddle (0, 2)

Solution steps

Find the critical points:

(0, 0), (0, 2)

\frac{\partial ^{2} f}{\partial x ^{2}} = - e^{- y} (- y^{2} + 4 y - x^{2} - 2)

D (x, y) = - 2 e^{- 2 y} x^{2} + 4 e^{- 2 y} + 2 e^{- 2 y} y^{2} - 8 e^{- 2 y} y

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(0, 2) :

Saddle

Minimum (0, 0), Saddle (0, 2)

What is the extreme f(x,y)=e^{-y}(x^2+y^2)+6 ?
The extreme f(x,y)=e^{-y}(x^2+y^2)+6 is Minimum(0,0),Saddle(0,2)