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

The extreme f(x,y)=x^2+xy+y^2+5y is Minimum(5/3 ,-10/3)

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

extreme points

f (x, y) = x^{2} + x y + y^{2} + 5 y

Minimum (\frac{5}{3}, - \frac{10}{3})

Solution steps

Find the critical points:

(\frac{5}{3}, - \frac{10}{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{5}{3}, - \frac{10}{3}) :

Minimum

Minimum (\frac{5}{3}, - \frac{10}{3})

f (x) = x^{2} + 5 x + 9

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

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

f (x, y) = 6 x^{2} - x y + 4 y^{2}

- x^{4} + 4 x^{3} + 8 x^{2}

What is the extreme f(x,y)=x^2+xy+y^2+5y ?
The extreme f(x,y)=x^2+xy+y^2+5y is Minimum(5/3 ,-10/3)