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

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

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

extreme points

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

Saddle (\frac{1}{2}, 0), Minimum (- \frac{1}{2}, 0), Maximum (\frac{1}{2}, \frac{1}{2}), Maximum (\frac{1}{2}, - \frac{1}{2})

Solution steps

Find the critical points:

(\frac{1}{2}, 0), (- \frac{1}{2}, 0), (\frac{1}{2}, \frac{1}{2}), (\frac{1}{2}, - \frac{1}{2})

\frac{\partial ^{2} f}{\partial x ^{2}} = 2 (2 e^{- x^{2} - y^{2}} y^{4} + 2 e^{- x^{2} - y^{2}} x y^{2} - 5 e^{- x^{2} - y^{2}} y^{2} - e^{- x^{2} - y^{2}} x + e^{- x^{2} - y^{2}})

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

Check the sign of at each critical point

Check critical point

(\frac{1}{2}, 0) :

Saddle

Check critical point

(- \frac{1}{2}, 0) :

Minimum

Check critical point

(\frac{1}{2}, \frac{1}{2}) :

Maximum

Check critical point

(\frac{1}{2}, - \frac{1}{2}) :

Maximum

Saddle (\frac{1}{2}, 0), Minimum (- \frac{1}{2}, 0), Maximum (\frac{1}{2}, \frac{1}{2}), Maximum (\frac{1}{2}, - \frac{1}{2})

f (x) = \frac{3 x ^{2}}{x ^{2} + 3}

\frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} - 4 x + 8

f (x, y) = x^{3} + 9 y^{4} - 20 x^{2} y + 2

2 x^{3} + 3 x^{2} + 4

f (x) = x^{2} + 2 y^{2} - 4 x + 8 y

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