What is the extreme f(x,y)=x^2+y^2+x^2y+9 ?

The extreme f(x,y)=x^2+y^2+x^2y+9 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^2y+9

extreme points

f (x, y) = x^{2} + y^{2} + x^{2} y + 9

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, y) = 2 x^{3} + x y^{2} + 5 x^{2} + y^{2} + 7

f (x) = - x^{2} + 2 x + 4, - 2 \leq x \leq 4

x^{\frac{2}{3}} (x - 5)

x^{\frac{2}{3}} (x - 2)

f (x) = 100 x (2 x + 3) (x - 5)

What is the extreme f(x,y)=x^2+y^2+x^2y+9 ?
The extreme f(x,y)=x^2+y^2+x^2y+9 is Minimum(0,0),Saddle(sqrt(2),-1),Saddle(-sqrt(2),-1)