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) = x^{3} - 3 x, 0 \leq x \leq 2

2 x - x

f (x) = 2 sin (x) - x

f (x) = x^{9} - 3 x^{7} + 2

ln^{2} (x^{2} + x + 1)

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)