What is the extreme f(x,y)=2x^2+xy^2-6xy+5x+2 ?

The extreme f(x,y)=2x^2+xy^2-6xy+5x+2 is Saddle(0,5),Saddle(0,1),Minimum(1,3)

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extreme f(x,y)=2x^2+xy^2-6xy+5x+2

extreme points

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

Saddle (0, 5), Saddle (0, 1), Minimum (1, 3)

Solution steps

Find the critical points:

(0, 5), (0, 1), (1, 3)

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

D (x, y) = 8 x - 4 y^{2} + 24 y - 36

Check the sign of at each critical point

Check critical point

(0, 5) :

Saddle

Check critical point

(0, 1) :

Saddle

Check critical point

(1, 3) :

Minimum

Saddle (0, 5), Saddle (0, 1), Minimum (1, 3)

What is the extreme f(x,y)=2x^2+xy^2-6xy+5x+2 ?
The extreme f(x,y)=2x^2+xy^2-6xy+5x+2 is Saddle(0,5),Saddle(0,1),Minimum(1,3)