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

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

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

extreme points

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

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 (2 e^{- x^{2} + 3 - y^{2}} x^{2} y^{2} + 4 e^{- x^{2} + 3 - y^{2}} y^{4} - 10 e^{- x^{2} + 3 - y^{2}} y^{2} - e^{- x^{2} + 3 - y^{2}} x^{2} + 2 e^{- x^{2} + 3 - y^{2}})

D (x, y) = 4 (2 e^{- x^{2} + 3 - y^{2}} x^{4} + 4 e^{- x^{2} + 3 - y^{2}} x^{2} y^{2} - 5 e^{- x^{2} + 3 - y^{2}} x^{2} - 2 e^{- x^{2} + 3 - y^{2}} y^{2} + e^{- x^{2} + 3 - y^{2}}) (2 e^{- x^{2} + 3 - y^{2}} x^{2} y^{2} + 4 e^{- x^{2} + 3 - y^{2}} y^{4} - 10 e^{- x^{2} + 3 - y^{2}} y^{2} - e^{- x^{2} + 3 - y^{2}} x^{2} + 2 e^{- x^{2} + 3 - y^{2}}) - 4 x^{2} (2 e^{- x^{2} + 3 - y^{2}} x^{2} y + 4 e^{- x^{2} + 3 - y^{2}} y^{3} - 6 e^{- x^{2} + 3 - y^{2}} 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)

p (x) = 12 x^{4} - 62 x^{3} + a x^{2} - 98 x + 30

y = 3 z^{2} - 3 x^{2}

4 - y^{3} - x^{2} - 3 x y

\frac{10 x ^{3} - 2}{2 x ^{3} + 5 x ^{2} + 9 x}

- 6 x^{2} + 336 x - 4320

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