What is the extreme f(x,y)=(4x^2+y^2)e^{-4y^2-x^2} ?

The extreme f(x,y)=(4x^2+y^2)e^{-4y^2-x^2} is Minimum(0,0),Maximum(1,0),Maximum(-1,0)

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

extreme points

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

Minimum (0, 0), Maximum (1, 0), Maximum (- 1, 0)

Solution steps

Find the critical points:

(0, 0), (1, 0), (- 1, 0)

\frac{\partial ^{2} f}{\partial x ^{2}} = 2 (32 e^{- x^{2} - 4 y^{2}} y^{4} + 128 e^{- x^{2} - 4 y^{2}} x^{2} y^{2} - 20 e^{- x^{2} - 4 y^{2}} y^{2} - 16 e^{- x^{2} - 4 y^{2}} x^{2} + e^{- x^{2} - 4 y^{2}})

D (x, y) = 4 (8 e^{- x^{2} - 4 y^{2}} x^{4} + 2 e^{- x^{2} - 4 y^{2}} x^{2} y^{2} - 20 e^{- x^{2} - 4 y^{2}} x^{2} - e^{- x^{2} - 4 y^{2}} y^{2} + 4 e^{- x^{2} - 4 y^{2}}) (32 e^{- x^{2} - 4 y^{2}} y^{4} + 128 e^{- x^{2} - 4 y^{2}} x^{2} y^{2} - 20 e^{- x^{2} - 4 y^{2}} y^{2} - 16 e^{- x^{2} - 4 y^{2}} x^{2} + e^{- x^{2} - 4 y^{2}}) - 4 x^{2} (8 e^{- x^{2} - 4 y^{2}} y^{3} + 32 e^{- x^{2} - 4 y^{2}} x^{2} y - 34 e^{- x^{2} - 4 y^{2}} y)^{2}

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(1, 0) :

Maximum

Check critical point

(- 1, 0) :

Maximum

Minimum (0, 0), Maximum (1, 0), Maximum (- 1, 0)

f (x) = x^{3} + 3 x^{2} - 9 x - 7

f (x) = x^{3} + 3 x^{2} - 9 x - 2

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

f (x) = 3 x + \frac{12}{x}

f (x, y) = 5 x y

What is the extreme f(x,y)=(4x^2+y^2)e^{-4y^2-x^2} ?
The extreme f(x,y)=(4x^2+y^2)e^{-4y^2-x^2} is Minimum(0,0),Maximum(1,0),Maximum(-1,0)