What is the extreme f(x,y)=-x^3+4x*y-2y^2+1 ?

The extreme f(x,y)=-x^3+4x*y-2y^2+1 is Saddle(0,0),Maximum(4/3 , 4/3)

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extreme f(x,y)=-x^3+4x*y-2y^2+1

extreme

Saddle (0, 0), Maximum (\frac{4}{3}, \frac{4}{3})

Solution steps

Find the critical points:

(0, 0), (\frac{4}{3}, \frac{4}{3})

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

D (x, y) = 24 x - 16

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(\frac{4}{3}, \frac{4}{3}) :

Maximum

Saddle (0, 0), Maximum (\frac{4}{3}, \frac{4}{3})

What is the extreme f(x,y)=-x^3+4x*y-2y^2+1 ?
The extreme f(x,y)=-x^3+4x*y-2y^2+1 is Saddle(0,0),Maximum(4/3 , 4/3)