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

The extreme f(x,y)=e^{x^2+xy+y^2} is Minimum(0,0)

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extreme f(x,y)=e^{x^2+xy+y^2}

extreme points

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

Minimum (0, 0)

Solution steps

Find the critical points:

(0, 0)

\frac{\partial ^{2} f}{\partial x ^{2}} = e^{x^{2} + x y + y^{2}} ((x + 2 y)^{2} + 2)

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

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Minimum (0, 0)

f (x) = - x^{2} + 10

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

f (x) = \frac{x \cdot e ^{x}}{e ^{x} - e}

f (x) = - x^{2} + 5 x - 6

f (x) = - x^{2} + 5 x - 1

What is the extreme f(x,y)=e^{x^2+xy+y^2} ?
The extreme f(x,y)=e^{x^2+xy+y^2} is Minimum(0,0)