What is the extreme f(x,y)=x^3y+12yx^2-8y ?

The extreme f(x,y)=x^3y+12yx^2-8y is Saddle(0.79085…,0),Saddle(-0.84693…,0),Saddle(-11.94392…,0)

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extreme f(x,y)=x^3y+12yx^2-8y

extreme

Saddle (0.79085 \dots, 0), Saddle (- 0.84693 \dots, 0), Saddle (- 11.94392 \dots, 0)

Solution steps

Find the critical points:

(0.79085 \dots, 0), (- 0.84693 \dots, 0), (- 11.94392 \dots, 0)

\frac{\partial ^{2} f}{\partial x ^{2}} = 0

D (x, y) = - 9 x^{4} - 144 x^{3} - 576 x^{2}

Check the sign of at each critical point

Check critical point

(0.79085 \dots, 0) :

Saddle

Check critical point

(- 0.84693 \dots, 0) :

Saddle

Check critical point

(- 11.94392 \dots, 0) :

Saddle

Saddle (0.79085 \dots, 0), Saddle (- 0.84693 \dots, 0), Saddle (- 11.94392 \dots, 0)

What is the extreme f(x,y)=x^3y+12yx^2-8y ?
The extreme f(x,y)=x^3y+12yx^2-8y is Saddle(0.79085…,0),Saddle(-0.84693…,0),Saddle(-11.94392…,0)