What is the extreme f(x,y)=x^3-y^2-4y+x^2y ?

The extreme f(x,y)=x^3-y^2-4y+x^2y is Maximum(0,-2),Saddle(1,-3/2),Saddle(-4,6)

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extreme f(x,y)=x^3-y^2-4y+x^2y

extreme

Maximum (0, - 2), Saddle (1, - \frac{3}{2}), Saddle (- 4, 6)

Solution steps

Find the critical points:

(0, - 2), (1, - \frac{3}{2}), (- 4, 6)

\frac{\partial ^{2} f}{\partial x ^{2}} = - 2

D (x, y) = - 2 (6 x + 2 y) - 4 x^{2}

Check the sign of at each critical point

Check critical point

(0, - 2) :

Maximum

Check critical point

(1, - \frac{3}{2}) :

Saddle

Check critical point

(- 4, 6) :

Saddle

Maximum (0, - 2), Saddle (1, - \frac{3}{2}), Saddle (- 4, 6)

What is the extreme f(x,y)=x^3-y^2-4y+x^2y ?
The extreme f(x,y)=x^3-y^2-4y+x^2y is Maximum(0,-2),Saddle(1,-3/2),Saddle(-4,6)