Test of Extrema
(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 = 33e-3 = 27e-3 [∴ y x3e-xgiven]
