What is the extreme f(x,y)=2x^4-x^2+10y^2 ?

The extreme f(x,y)=2x^4-x^2+10y^2 is Saddle(0,0),Minimum(-1/2 ,0),Minimum(1/2 ,0)

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extreme f(x,y)=2x^4-x^2+10y^2

extreme points

f (x, y) = 2 x^{4} - x^{2} + 10 y^{2}

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

Solution steps

Find the critical points:

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

\frac{\partial ^{2} f}{\partial x ^{2}} = 20

D (x, y) = 20 (24 x^{2} - 2)

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

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

Minimum

Check critical point

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

Minimum

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

f (x, y) = x^{3} - y^{3} - 6 x y - 4

f (x) = x^{3} - 3 x^{2} - 9 x + 15

f (x) = x^{4} - 8 x^{3} + 3

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

x^{\frac{1}{3}} (x + 4)

What is the extreme f(x,y)=2x^4-x^2+10y^2 ?
The extreme f(x,y)=2x^4-x^2+10y^2 is Saddle(0,0),Minimum(-1/2 ,0),Minimum(1/2 ,0)