What is the extreme f(x)=x^4+y^4-4xy+1 ?

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

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extreme f(x)=x^4+y^4-4xy+1

extreme points

f (x) = x^{4} + y^{4} - 4 x y + 1

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

Solution steps

Find the critical points:

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

\frac{\partial ^{2} f}{\partial x ^{2}} = 12 y^{2}

D (x, y) = 144 x^{2} y^{2} - 16

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(- 1, - 1) :

Minimum

Check critical point

(1, 1) :

Minimum

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

f (x) = \frac{3 x - 6}{x + 2}

f (x, y) = 1 - x^{2} + 2 y^{2}

f (x, y) = 60 x - 2 x^{2} - 3 y^{2} + 30 y

\frac{x ^{2}}{x - 4}

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

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