What is the extreme f(x)=((x^2+y^2)e^{(-x)/3}) ?

The extreme f(x)=((x^2+y^2)e^{(-x)/3}) is Minimum(0,0),Saddle(6,0)

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extreme f(x)=((x^2+y^2)e^{(-x)/3})

extreme

Minimum (0, 0), Saddle (6, 0)

Solution steps

Find the critical points:

(0, 0), (6, 0)

\frac{\partial ^{2} f}{\partial x ^{2}} = 2 e^{- \frac{x}{3}}

D (x, y) = \frac{2 e ^{- \frac{2 x}{3}} x ^{2} - 24 e ^{- \frac{2 x}{3}} x - 2 e ^{- \frac{2 x}{3}} y ^{2}}{9} + 4 e^{- \frac{2 x}{3}}

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(6, 0) :

Saddle

Minimum (0, 0), Saddle (6, 0)

What is the extreme f(x)=((x^2+y^2)e^{(-x)/3}) ?
The extreme f(x)=((x^2+y^2)e^{(-x)/3}) is Minimum(0,0),Saddle(6,0)