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

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

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

extreme

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

Solution steps

Find the critical points:

(0, 0), (2, 0)

\frac{\partial ^{2} f}{\partial x ^{2}} = 2 e^{- x}

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

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(2, 0) :

Saddle

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

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