What is the extreme f(x,y)=y^2(x-1)^2+x^2(y+4)^2 ?

The extreme f(x,y)=y^2(x-1)^2+x^2(y+4)^2 is Minimum(0,0),Saddle(1/2 ,-2),Minimum(1,-4)

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extreme f(x,y)=y^2(x-1)^2+x^2(y+4)^2

extreme points

f (x, y) = y^{2} (x - 1)^{2} + x^{2} (y + 4)^{2}

Minimum (0, 0), Saddle (\frac{1}{2}, - 2), Minimum (1, - 4)

Solution steps

Find the critical points:

(0, 0), (\frac{1}{2}, - 2), (1, - 4)

\frac{\partial ^{2} f}{\partial x ^{2}} = 4 x^{2} - 4 x + 2

D (x, y) = - 48 x^{2} y^{2} - 192 x^{2} y - 128 x^{2} + 48 x y^{2} + 64 x y - 128 x - 8 y^{2} + 32 y + 64

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(\frac{1}{2}, - 2) :

Saddle

Check critical point

(1, - 4) :

Minimum

Minimum (0, 0), Saddle (\frac{1}{2}, - 2), Minimum (1, - 4)

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

2 x^{3} - 3 x^{2} - 12 x + 1

f (x) = 324 x - 72 x^{2} + 4 x^{3}

f (x) = 2 x + 3 y

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

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