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

The extreme f(x,y)=x^2+xy+y^2+y is Minimum(1/3 ,-2/3)

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

extreme points

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

Minimum (\frac{1}{3}, - \frac{2}{3})

Solution steps

Find the critical points:

(\frac{1}{3}, - \frac{2}{3})

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

D (x, y) = 3

Check the sign of at each critical point

Check critical point

(\frac{1}{3}, - \frac{2}{3}) :

Minimum

Minimum (\frac{1}{3}, - \frac{2}{3})

What is the extreme f(x,y)=x^2+xy+y^2+y ?
The extreme f(x,y)=x^2+xy+y^2+y is Minimum(1/3 ,-2/3)