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

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

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

extreme points

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

Saddle (0, 0), Maximum (- 4, - 2), Saddle (0, 0)

Solution steps

Find the critical points:

(0, 0), (- 4, - 2), (0, 0)

\frac{\partial ^{2} f}{\partial x ^{2}} = e^{- y + x} x^{2} - 2 e^{- y + x} y^{2} + 8 e^{- y + x} y - 4 e^{- y + x}

D (x, y) = e^{- y + x} (x^{2} + 4 x - 2 y^{2} + 2) (e^{- y + x} x^{2} - 2 e^{- y + x} y^{2} + 8 e^{- y + x} y - 4 e^{- y + x}) - e^{2 (- y + x)} (x^{2} - 2 y^{2} + 2 x + 4 y)^{2}

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(- 4, - 2) :

Maximum

Check critical point

(0, 0) :

Saddle

Saddle (0, 0), Maximum (- 4, - 2), Saddle (0, 0)

f (x) = \frac{( 1 + x ^{2} )}{4 - x ^{2}}

f (x) = x + \frac{1}{x ^{2}}

f (x) = \frac{x ^{2}}{( x ^{2} - 9 )}

f (x) = \frac{x ^{2} + 9}{x}

g (x, y) = x - c + y

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