What is the extreme f(x,y)=x^{(2)}+y^{(2)}+x^{(2)}y+4 ?

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

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

extreme points

f (x, y) = 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)

2 cos (x) + 2 sin (x)

f (x) = \frac{8}{x ^{2} - 4}

\frac{3 ( x - 7 ) ( x - 6 )}{3 ( x - 7 )}

f (x) = x^{3} + x + 1

f (x) = 40 x - 3 x^{2}

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