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})

F (x, y) = x^{3} ln (y^{2} + 1) - 2 y

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

2 x + 10

f (x, y) = e^{- x^{2} - y^{2}}

f (x, y) = 72 y^{2} + x^{2} - x^{2} y

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)