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

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

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

extreme points

f (x, y) = x^{2} + x y + \frac{1}{2} y^{2} - 2 x + y

Minimum (3, - 4)

Solution steps

Find the critical points:

(3, - 4)

\frac{\partial ^{2} f}{\partial x ^{2}} = 1

D (x, y) = 1

Check the sign of at each critical point

Check critical point

(3, - 4) :

Minimum

Minimum (3, - 4)

f (x) = 14 x^{2} - 2 x^{3} 2 y^{2} + 4 x y

2 x^{2} + y^{2} + x y - 3 x

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

f (x) = x^{4} - 72 x^{2}

f (x) = - 3 x^{4} ln (3 x), (0, 4)

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