What is the extreme (1/2-x^2+y^2)e^{1-x^2-y^2} ?

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

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extreme (1/2-x^2+y^2)e^{1-x^2-y^2}

extreme points

(\frac{1}{2} - x^{2} + y^{2}) e^{1 - x^{2} - y^{2}}

Saddle (0, 0), Minimum (\frac{3}{2}, 0), Minimum (- \frac{3}{2}, 0)

Solution steps

Find the critical points:

(0, 0), (\frac{3}{2}, 0), (- \frac{3}{2}, 0)

\frac{\partial ^{2} f}{\partial x ^{2}} = - 4 e^{- x^{2} + 1 - y^{2}} x^{2} y^{2} + 4 e^{- x^{2} + 1 - y^{2}} y^{4} - 8 e^{- x^{2} + 1 - y^{2}} y^{2} + 2 e^{- x^{2} + 1 - y^{2}} x^{2} + e^{- x^{2} + 1 - y^{2}}

D (x, y) = (- 4 e^{- x^{2} + 1 - y^{2}} x^{4} + 4 e^{- x^{2} + 1 - y^{2}} x^{2} y^{2} + 12 e^{- x^{2} + 1 - y^{2}} x^{2} - 2 e^{- x^{2} + 1 - y^{2}} y^{2} - 3 e^{- x^{2} + 1 - y^{2}}) (- 4 e^{- x^{2} + 1 - y^{2}} x^{2} y^{2} + 4 e^{- x^{2} + 1 - y^{2}} y^{4} - 8 e^{- x^{2} + 1 - y^{2}} y^{2} + 2 e^{- x^{2} + 1 - y^{2}} x^{2} + e^{- x^{2} + 1 - y^{2}}) - 4 x^{2} (2 e^{- x^{2} + 1 - y^{2}} x^{2} y - 2 e^{- x^{2} + 1 - y^{2}} y^{3} - e^{- x^{2} + 1 - y^{2}} y)^{2}

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(\frac{3}{2}, 0) :

Minimum

Check critical point

(- \frac{3}{2}, 0) :

Minimum

Saddle (0, 0), Minimum (\frac{3}{2}, 0), Minimum (- \frac{3}{2}, 0)

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