(a) If X** = **∞

When x is slightly less than ∞. dy/dx = (+)(+)(-1) = (-)

When x is slightly more than ∞. dy/dx = (+)(+)(-1) = (-)

The above test shows that dy/dx does not change sign when x =∞. Hence, there is no minima nor maxima of the function at x = ∞.

(b) If x = 0

When x is slightly less than 0, dy/dx = (+)(+)(+) = (+)

When x is slightly more than 0, dy/dx = (+)(+)(+) = (+)

The above test shows that dy/dx does not change the sign when x = 0. Hence, x = 0, gives neither a maximum not a minimum value of the function.

(c) If x = 3

When x is slightly less than 3, dy/dx = (+) (+) (+) = (+)

When x is slightly more than 3, dy/dx = (+) (+) (-) = (-)

The above test reveals that dy/dx changes sign from + ve to – ve, when x = 3. Therefore x =3 gives maximum to the function and the same maximum value is given by

Max. *y = *3^{3}e^{-3} = 27e^{-3} [∴ y x^{3}e^{-x}given]