What is the extreme f(x,y)=(x^2+3y^2)e^{1-x^2-y^2} ?

The extreme f(x,y)=(x^2+3y^2)e^{1-x^2-y^2} is Minimum(0,0),Saddle(1,0),Saddle(-1,0)

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extreme f(x,y)=(x^2+3y^2)e^{1-x^2-y^2}

extreme

Minimum (0, 0), Saddle (1, 0), Saddle (- 1, 0)

Solution steps

Find the critical points:

(0, 0), (1, 0), (- 1, 0)

\frac{\partial ^{2} f}{\partial x ^{2}} = 2 (6 e^{- x^{2} - y^{2} + 1} y^{4} + 2 e^{- x^{2} - y^{2} + 1} x^{2} y^{2} - 15 e^{- x^{2} - y^{2} + 1} y^{2} - e^{- x^{2} - y^{2} + 1} x^{2} + 3 e^{- x^{2} - y^{2} + 1})

D (x, y) = 4 (2 e^{- x^{2} - y^{2} + 1} x^{4} + 6 e^{- x^{2} - y^{2} + 1} x^{2} y^{2} - 5 e^{- x^{2} - y^{2} + 1} x^{2} - 3 e^{- x^{2} - y^{2} + 1} y^{2} + e^{- x^{2} - y^{2} + 1}) (6 e^{- x^{2} - y^{2} + 1} y^{4} + 2 e^{- x^{2} - y^{2} + 1} x^{2} y^{2} - 15 e^{- x^{2} - y^{2} + 1} y^{2} - e^{- x^{2} - y^{2} + 1} x^{2} + 3 e^{- x^{2} - y^{2} + 1}) - 4 x^{2} (6 e^{- x^{2} - y^{2} + 1} y^{3} + 2 e^{- x^{2} - y^{2} + 1} x^{2} y - 8 e^{- x^{2} - y^{2} + 1} y)^{2}

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(1, 0) :

Saddle

Check critical point

(- 1, 0) :

Saddle

Minimum (0, 0), Saddle (1, 0), Saddle (- 1, 0)

What is the extreme f(x,y)=(x^2+3y^2)e^{1-x^2-y^2} ?
The extreme f(x,y)=(x^2+3y^2)e^{1-x^2-y^2} is Minimum(0,0),Saddle(1,0),Saddle(-1,0)