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

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

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

extreme points

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

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

Solution steps

Find the critical points:

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

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

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

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(2, 4) :

Maximum

Check critical point

(0, 0) :

Saddle

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

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

\frac{d}{d x} (x y^{2})

f (x) = x^{3} - x^{2} - x - 10

y = 2 x^{3} + 3 x^{2} - 36 x + 4

f (x, y) = - 2 x^{2} + 5 x y - 3 y^{2} + x + y

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