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

The extreme f(x,y)=xy-x^3-y^2 is Saddle(0,0),Maximum(1/6 , 1/12)

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

extreme points

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

Saddle (0, 0), Maximum (\frac{1}{6}, \frac{1}{12})

Solution steps

Find the critical points:

(0, 0), (\frac{1}{6}, \frac{1}{12})

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

D (x, y) = 12 x - 1

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(\frac{1}{6}, \frac{1}{12}) :

Maximum

Saddle (0, 0), Maximum (\frac{1}{6}, \frac{1}{12})

g (x, y) = x y

f (x, y) = x^{3} + y^{3} - 3 x - 12 y + 20

f (x) = x^{3} - 6 x^{2} + 5, - 3 \leq x \leq 5

t^{3} - 5 t^{2} - 2

G (b, a) = - b + a

What is the extreme f(x,y)=xy-x^3-y^2 ?
The extreme f(x,y)=xy-x^3-y^2 is Saddle(0,0),Maximum(1/6 , 1/12)