What is the extreme x^4+y^4-4xy+5 ?

The extreme x^4+y^4-4xy+5 is Saddle(0,0),Minimum(-1,-1),Minimum(1,1)

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extreme x^4+y^4-4xy+5

extreme points

x^{4} + y^{4} - 4 x y + 5

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, y) = y x^{2} - 2 x^{2} - y^{2}

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

f (x) = x^{2} + 6 x + 9

f (x) = x + 5

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

What is the extreme x^4+y^4-4xy+5 ?
The extreme x^4+y^4-4xy+5 is Saddle(0,0),Minimum(-1,-1),Minimum(1,1)