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)

f (x) = x 36 - x^{2}

f (x) = x^{4} + \frac{4 x ^{3}}{3} - 4 x^{2}

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

f (x) = 3 x^{3} - 6 x^{2} - 5 x + 1

- x^{3} + 3 x^{2} - 2

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)