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

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

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

extreme points

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

Minimum (0, 0), Saddle (- 1, 2), Saddle (- 1, - 2)

Solution steps

Find the critical points:

(0, 0), (- 1, 2), (- 1, - 2)

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

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

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(- 1, 2) :

Saddle

Check critical point

(- 1, - 2) :

Saddle

Minimum (0, 0), Saddle (- 1, 2), Saddle (- 1, - 2)

f (x) = - 16 + 4 ln (x)

f (x, y) = x^{2} - 2 y^{2} + x y - 3 x + 3 y + 4

f (x) = \frac{x ^{3}}{3} + x^{2} - 8 x

- x^{4} y^{2} - x^{4} + 2 x^{3} y + 2 x^{2} y + x^{2} - 2 x - 2

f (x \cdot y) = x y + 8^{x y}

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