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

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

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

extreme points

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

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

\frac{ln ( 4 x )}{x}

f (x) = \frac{4}{x - 3}

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

f (x) = 19 + 4 x - x^{2}

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

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