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

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

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extreme f(x)=x^2+x^2y+y^2+1

extreme points

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

Minimum (0, 0), Saddle (2, - 1), Saddle (- 2, - 1)

Solution steps

Find the critical points:

(0, 0), (2, - 1), (- 2, - 1)

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

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

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(2, - 1) :

Saddle

Check critical point

(- 2, - 1) :

Saddle

Minimum (0, 0), Saddle (2, - 1), Saddle (- 2, - 1)

f (x) = 8 x - 9 x^{(\frac{8}{9})}

f (x) = 33 x + 66 y + x y - x^{2} - 3 y^{2}

f (x, y) = 3 x^{2} + 3 y^{2} + x^{3} + 4 y

f (x) = - 3 x^{3} - 10

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

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