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

The extreme F(x,y)=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,y)=x^2+x^2y+y^2+1

extreme points

F (x, y) = 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) = - \frac{5 x}{4 x ^{2} + 7}

f (x) = e^{- 1.5 x^{2}}

f (x) = x^{3} - 9 x^{2} + 24 x - 2

f (x, y) = 2 x^{2} + 3 x y + 4 y^{2} + 7 x + 11 y

f (x) = x^{4} - 5 x^{2}

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