What is the extreme f(x,y)=288y^2+x^2-x^2y ?

The extreme f(x,y)=288y^2+x^2-x^2y is Minimum(0,0),Saddle(24,1),Saddle(-24,1)

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extreme f(x,y)=288y^2+x^2-x^2y

extreme points

f (x, y) = 288 y^{2} + x^{2} - x^{2} y

Minimum (0, 0), Saddle (24, 1), Saddle (- 24, 1)

Solution steps

Find the critical points:

(0, 0), (24, 1), (- 24, 1)

\frac{\partial ^{2} f}{\partial x ^{2}} = 576

D (x, y) = 576 (2 - 2 y) - 4 x^{2}

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(24, 1) :

Saddle

Check critical point

(- 24, 1) :

Saddle

Minimum (0, 0), Saddle (24, 1), Saddle (- 24, 1)

f (x, y) = x (x - 4) y (y - 4)

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

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

f (x) = 5 (5 x)^{x}, 0.05 \leq x \leq 1

f (x) = 3 x^{3} e^{- x}, - 1 \leq x \leq 6

What is the extreme f(x,y)=288y^2+x^2-x^2y ?
The extreme f(x,y)=288y^2+x^2-x^2y is Minimum(0,0),Saddle(24,1),Saddle(-24,1)