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

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

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

extreme points

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

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

Solution steps

Find the critical points:

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

Minimum

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

f (x) = \frac{x ^{3}}{x ^{2} + 3}

f (x) = 2 x^{3} + x^{2} + 2 x

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

f (x) = x^{2} - 8 ln (x), 1 \leq x \leq 5

\frac{x + 5}{x ^{2} - 2 x - 35}

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