What is the extreme f(x,y)=5x^2+xy^2+y^2 ?

The extreme f(x,y)=5x^2+xy^2+y^2 is Minimum(0,0),Saddle(-1,sqrt(10)),Saddle(-1,-sqrt(10))

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extreme f(x,y)=5x^2+xy^2+y^2

extreme

Minimum (0, 0), Saddle (- 1, 10), Saddle (- 1, - 10)

Solution steps

Find the critical points:

(0, 0), (- 1, 10), (- 1, - 10)

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

D (x, y) = 10 (2 x + 2) - 4 y^{2}

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(- 1, 10) :

Saddle

Check critical point

(- 1, - 10) :

Saddle

Minimum (0, 0), Saddle (- 1, 10), Saddle (- 1, - 10)

What is the extreme f(x,y)=5x^2+xy^2+y^2 ?
The extreme f(x,y)=5x^2+xy^2+y^2 is Minimum(0,0),Saddle(-1,sqrt(10)),Saddle(-1,-sqrt(10))