What is the extreme F(x,y)=x^3-2x^2y^2-2x-4y+10 ?

The extreme F(x,y)=x^3-2x^2y^2-2x-4y+10 is Saddle(1.19960…,-0.69490…)

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extreme F(x,y)=x^3-2x^2y^2-2x-4y+10

extreme points

F (x, y) = x^{3} - 2 x^{2} y^{2} - 2 x - 4 y + 10

Saddle (1.19960 \dots, - 0.69490 \dots)

Solution steps

Find the critical points:

(1.19960 \dots, - 0.69490 \dots)

\frac{\partial ^{2} f}{\partial x ^{2}} = - 4 x^{2}

D (x, y) = - 24 x^{3} - 48 x^{2} y^{2}

Check the sign of at each critical point

Check critical point

(1.19960 \dots, - 0.69490 \dots) :

Saddle

Saddle (1.19960 \dots, - 0.69490 \dots)

(x + 3)^{4} (2 - x)

f (x, y) = x^{2} + x y + y^{2} + 8 y

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

f (x) = x^{2} + 5 x + 9

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

What is the extreme F(x,y)=x^3-2x^2y^2-2x-4y+10 ?
The extreme F(x,y)=x^3-2x^2y^2-2x-4y+10 is Saddle(1.19960…,-0.69490…)