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

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

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

extreme points

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

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

D (x, y) = 9 (4 e^{- x^{2} + y^{2}} x^{4} + 4 e^{- x^{2} + y^{2}} x^{2} y^{2} - 10 e^{- x^{2} + y^{2}} x^{2} - 2 e^{- x^{2} + y^{2}} y^{2} + 2 e^{- x^{2} + y^{2}}) (4 e^{- x^{2} + y^{2}} y^{4} + 4 e^{- x^{2} + y^{2}} x^{2} y^{2} + 10 e^{- x^{2} + y^{2}} y^{2} + 2 e^{- x^{2} + y^{2}} x^{2} + 2 e^{- x^{2} + y^{2}}) - 9 (- 4 e^{- x^{2} + y^{2}} x^{3} y - 4 e^{- x^{2} + y^{2}} x y^{3})^{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)=3(x^2+y^2)e^{y^2-x^2} ?
The extreme f(x,y)=3(x^2+y^2)e^{y^2-x^2} is Minimum(0,0),Saddle(-1,0),Saddle(1,0)