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

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

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

extreme points

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

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

Solution steps

Find the critical points:

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

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

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

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(- 1, 2) :

Saddle

Check critical point

(- 1, - 2) :

Saddle

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

f (t) = 10 e^{a t}

f (x) = 2 x^{3} - 6 x^{2} - 90 x + 1

f (x) = (11 - x) (x + 1)^{2}

f (x, y) = 400 - 16 x^{2} - 49 y^{2}

f (x) = 4 x^{3} + 9 x^{2} - 12 x + 5

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