What is the extreme f(x,y)=x^2-6xy+y^3+3x+6y-2 ?

The extreme f(x,y)=x^2-6xy+y^3+3x+6y-2 is Minimum(27/2 ,5),Saddle(3/2 ,1)

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extreme f(x,y)=x^2-6xy+y^3+3x+6y-2

extreme points

f (x, y) = x^{2} - 6 x y + y^{3} + 3 x + 6 y - 2

Minimum (\frac{27}{2}, 5), Saddle (\frac{3}{2}, 1)

Solution steps

Find the critical points:

(\frac{27}{2}, 5), (\frac{3}{2}, 1)

\frac{\partial ^{2} f}{\partial x ^{2}} = 6 y

D (x, y) = 12 y - 36

Check the sign of at each critical point

Check critical point

(\frac{27}{2}, 5) :

Minimum

Check critical point

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

Saddle

Minimum (\frac{27}{2}, 5), Saddle (\frac{3}{2}, 1)

f (x) = 3 x^{\frac{2}{3}}

f (x) = \frac{1}{x} \cdot ln^{2} (3 x)

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

f (x, y) = 2 x^{2} + y^{2} + 6 x y + 10 x - 6 y + 5

f (x, y) = 2 y^{3} - 3 y^{2} - 12 y + 2 x^{2} - 6 x + 3

What is the extreme f(x,y)=x^2-6xy+y^3+3x+6y-2 ?
The extreme f(x,y)=x^2-6xy+y^3+3x+6y-2 is Minimum(27/2 ,5),Saddle(3/2 ,1)