What is the extreme g(x,y)=x^2+y^2-xy+2y+1 ?

The extreme g(x,y)=x^2+y^2-xy+2y+1 is Minimum(-2/3 ,-4/3)

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extreme g(x,y)=x^2+y^2-xy+2y+1

extreme points

g (x, y) = x^{2} + y^{2} - x y + 2 y + 1

Minimum (- \frac{2}{3}, - \frac{4}{3})

Solution steps

Find the critical points:

(- \frac{2}{3}, - \frac{4}{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{2}{3}, - \frac{4}{3}) :

Minimum

Minimum (- \frac{2}{3}, - \frac{4}{3})

f (x, y) = (x - y) (1 - x y)

f (x, y) = 3 x - x^{3} - 2 y^{2} + y^{4}

f (x) = x^{4} - 2 x^{3} + x + 1

f (x) = 3 x^{3} + 16 x

What is the extreme g(x,y)=x^2+y^2-xy+2y+1 ?
The extreme g(x,y)=x^2+y^2-xy+2y+1 is Minimum(-2/3 ,-4/3)