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

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

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

extreme points

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

Minimum (0, 1), Saddle (- \frac{1}{2}, 1), Saddle (\frac{1}{2}, 1)

Solution steps

Find the critical points:

(0, 1), (- \frac{1}{2}, 1), (\frac{1}{2}, 1)

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

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

Check the sign of at each critical point

Check critical point

(0, 1) :

Minimum

Check critical point

(- \frac{1}{2}, 1) :

Saddle

Check critical point

(\frac{1}{2}, 1) :

Saddle

Minimum (0, 1), Saddle (- \frac{1}{2}, 1), Saddle (\frac{1}{2}, 1)

f (x) = \frac{x}{x ^{2} - x + 1}, 0 \leq x \leq 3

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

F (I, J) = 25 I N + 12 J N

(x - x^{2})^{2}, - 1 \leq x \leq 1

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

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