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 points

f (x) = ((x^{2} + y^{2}) e^{\frac{- x}{3}})

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)

f (x) = 7 x + ln (x)

y = x^{\frac{2}{7}} (x^{2} - 5)

y = - t u (t - 1) + t u (t - 2)

f (x) = 2 x^{2} + 2 y^{2} - 8 x + 16 y

f (x) = x^{2} + y^{2} - 8 y + 16

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)