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

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

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

extreme points

f (x, y) = x^{2} + y^{3} - 6 x y + 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) = 2 x^{3} + 9 x^{2} - 24 x

f (x, y) = 2 x^{2} - 4 x + y^{2} - 8 y + 1

f (x) = \frac{x ^{3}}{3} - 6 x^{2}

f (x) = - x^{3} + 12 x - 14

f (x) = 200 - 8 y = 0

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