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

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

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

extreme points

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

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

Solution steps

Find the critical points:

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

Minimum

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

e^{x}

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

\frac{24}{r} + 2 π r^{2}

\frac{d}{d x} (9 x^{3} y^{5})

f (x) = 6 y^{2} + 4 x^{2} - 12 x y - 4 y

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