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

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

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

extreme points

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

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)

f (x, y) = x^{3} - 3 x

f (x) = 2 x^{2} + 3, - 1 \leq x \leq 2

f (x) = sin (π x) (- \frac{1}{3}, \frac{1}{3})

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

f (x, y) = 3 x^{2} - 6 x + 4 y^{2} + 8

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