What is the extreme x^2+y^2+x^2y+4 ?

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

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extreme x^2+y^2+x^2y+4

extreme points

x^{2} + y^{2} + x^{2} y + 4

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 y + 2) - 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) = 4 x^{5} - 8 x^{4} - 5 x^{3} + 10 x^{2} + x - 2

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

f (x, y) = 50 y^{2} + x^{2} - x^{2} y

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

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

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