What is the extreme f(x,y)=5x^4-x^2+2y^2 ?

The extreme f(x,y)=5x^4-x^2+2y^2 is Saddle(0,0),Minimum(-1/(sqrt(10)),0),Minimum(1/(sqrt(10)),0)

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extreme f(x,y)=5x^4-x^2+2y^2

extreme points

f (x, y) = 5 x^{4} - x^{2} + 2 y^{2}

Saddle (0, 0), Minimum (- \frac{1}{10}, 0), Minimum (\frac{1}{10}, 0)

Solution steps

Find the critical points:

(0, 0), (- \frac{1}{10}, 0), (\frac{1}{10}, 0)

\frac{\partial ^{2} f}{\partial x ^{2}} = 4

D (x, y) = 4 (60 x^{2} - 2)

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(- \frac{1}{10}, 0) :

Minimum

Check critical point

(\frac{1}{10}, 0) :

Minimum

Saddle (0, 0), Minimum (- \frac{1}{10}, 0), Minimum (\frac{1}{10}, 0)

f (x) = - \frac{1}{x ^{2}}, 0.5 \leq x \leq 2

f (x) = - \frac{1}{x ^{2}}, 0.5 \leq x \leq 4

f (x) = x^{2} - \frac{54}{x ^{2}}

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

f (x, y) = x^{2} - 2 x + y^{2} - 6 y + 30

What is the extreme f(x,y)=5x^4-x^2+2y^2 ?
The extreme f(x,y)=5x^4-x^2+2y^2 is Saddle(0,0),Minimum(-1/(sqrt(10)),0),Minimum(1/(sqrt(10)),0)