What is the extreme f(x,y)=12xy^2-16x+25xy^3-18x^2y^3 ?

The extreme f(x,y)=12xy^2-16x+25xy^3-18x^2y^3 is Saddle(0.90503…,-0.91854…),Saddle(0,0.72790…)

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extreme f(x,y)=12xy^2-16x+25xy^3-18x^2y^3

extreme points

f (x, y) = 12 x y^{2} - 16 x + 25 x y^{3} - 18 x^{2} y^{3}

Saddle (0.90503 \dots, - 0.91854 \dots), Saddle (0, 0.72790 \dots)

Solution steps

Find the critical points:

(0.90503 \dots, - 0.91854 \dots), (0, 0.72790 \dots)

\frac{\partial ^{2} f}{\partial x ^{2}} = - 108 x^{2} y + 24 x + 150 x y

D (x, y) = - 7776 x^{2} y^{4} + 10800 x y^{4} - 5625 y^{4} + 4320 x y^{3} - 3600 y^{3} - 576 y^{2}

Check the sign of at each critical point

Check critical point

(0.90503 \dots, - 0.91854 \dots) :

Saddle

Check critical point

(0, 0.72790 \dots) :

Saddle

Saddle (0.90503 \dots, - 0.91854 \dots), Saddle (0, 0.72790 \dots)

f (x, y) = \frac{1}{3} x^{3} + \frac{1}{3} y^{3} - x y + 4

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

f (x) = \frac{1}{x - 2} - 3

T (x, y) = (x + 3 y, - x + 5 y)

f (x) = x (12 - 2 x)^{2}

What is the extreme f(x,y)=12xy^2-16x+25xy^3-18x^2y^3 ?
The extreme f(x,y)=12xy^2-16x+25xy^3-18x^2y^3 is Saddle(0.90503…,-0.91854…),Saddle(0,0.72790…)