What is the extreme f(x,y)=72x^4+y^2-24xy ?

The extreme f(x,y)=72x^4+y^2-24xy is Saddle(0,0),Minimum(-1,-12),Minimum(1,12)

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extreme f(x,y)=72x^4+y^2-24xy

extreme points

f (x, y) = 72 x^{4} + y^{2} - 24 x y

Saddle (0, 0), Minimum (- 1, - 12), Minimum (1, 12)

Solution steps

Find the critical points:

(0, 0), (- 1, - 12), (1, 12)

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

D (x, y) = 1728 x^{2} - 576

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(- 1, - 12) :

Minimum

Check critical point

(1, 12) :

Minimum

Saddle (0, 0), Minimum (- 1, - 12), Minimum (1, 12)

f (x) = x^{3} + y^{3} - 18 x y

f (x) = x^{3} - 6 x^{2} + 9 x - 1

f (x) = x^{3} - 6 x^{2} + 9 x - 4

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

f (x) = \frac{( 3 x - 57 )}{( x - 85 ) ^{7}}

What is the extreme f(x,y)=72x^4+y^2-24xy ?
The extreme f(x,y)=72x^4+y^2-24xy is Saddle(0,0),Minimum(-1,-12),Minimum(1,12)