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

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

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

extreme points

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

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

f (x) = x^{3} - 9 x^{2} + 24 x - 7

f (x) = \frac{1}{x ^{3} - 6 x ^{2} + 1}

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

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