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

The extreme f(x,y)=e^x(x^2+y^2) is Minimum(0,0),Saddle(-2,0)

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

extreme points

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

Minimum (0, 0), Saddle (- 2, 0)

Solution steps

Find the critical points:

(0, 0), (- 2, 0)

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

D (x, y) = 2 e^{2 x} x^{2} + 8 e^{2 x} x - 2 e^{2 x} y^{2} + 4 e^{2 x}

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(- 2, 0) :

Saddle

Minimum (0, 0), Saddle (- 2, 0)

f (x, y) = x^{3} - 2 y^{2} - 6 x y + 4

f (x) = 2 x^{3} + x^{2} + 8 x

f (x) = 2 x^{3} - 6 x y + 3 y^{2}

f (x) = 168000 x - 100 x^{2}

f (x) = x^{2} (6 - x)^{3}

What is the extreme f(x,y)=e^x(x^2+y^2) ?
The extreme f(x,y)=e^x(x^2+y^2) is Minimum(0,0),Saddle(-2,0)