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

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

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

extreme points

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

Saddle (0, 0), Minimum (- \frac{1}{4}, - \frac{1}{4}), Minimum (\frac{1}{4}, \frac{1}{4})

Solution steps

Find the critical points:

(0, 0), (- \frac{1}{4}, - \frac{1}{4}), (\frac{1}{4}, \frac{1}{4})

\frac{\partial ^{2} f}{\partial x ^{2}} = 48 y^{2}

D (x, y) = 2304 x^{2} y^{2} - 1

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(- \frac{1}{4}, - \frac{1}{4}) :

Minimum

Check critical point

(\frac{1}{4}, \frac{1}{4}) :

Minimum

Saddle (0, 0), Minimum (- \frac{1}{4}, - \frac{1}{4}), Minimum (\frac{1}{4}, \frac{1}{4})

f (x, y) = ln (x^{3} y + 3 x y^{3})

f (x) = 2 x ln (9 x)

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

f (x) = 4 x^{3} + 3 x^{2} - 60 x - 12

f (x) = (x + 5)^{3} + 7

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