What is the extreme f(x,y)=8y^2+x^2-x^2y ?

The extreme f(x,y)=8y^2+x^2-x^2y is Minimum(0,0),Saddle(4,1),Saddle(-4,1)

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extreme f(x,y)=8y^2+x^2-x^2y

extreme points

f (x, y) = 8 y^{2} + x^{2} - x^{2} y

Minimum (0, 0), Saddle (4, 1), Saddle (- 4, 1)

Solution steps

Find the critical points:

(0, 0), (4, 1), (- 4, 1)

\frac{\partial ^{2} f}{\partial x ^{2}} = 16

D (x, y) = 16 (2 - 2 y) - 4 x^{2}

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(4, 1) :

Saddle

Check critical point

(- 4, 1) :

Saddle

Minimum (0, 0), Saddle (4, 1), Saddle (- 4, 1)

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

f (x) = x - 8 3 x

f (x) = 2 x^{3} - 15 x^{2} + 36 x + 10

f (x) = x^{3} - 12 x^{2} + 45 x + 5

f (x) = - x^{\frac{5}{3}} + 5 3 x

What is the extreme f(x,y)=8y^2+x^2-x^2y ?
The extreme f(x,y)=8y^2+x^2-x^2y is Minimum(0,0),Saddle(4,1),Saddle(-4,1)