Note (Geometric interpretation of point of inflection):
Let represent the distance covered by a moving body up to time . Then, represents the velocity of the body at time and represents the acceleration of the body at time . Then, a point is a point of inflection means that if the body was accelerating till then it starts decelerating from the time onwards, or vice-versa.
11.2.14
Remarks:
(i)
In general, the condition need not imply that the function has a point of inflection at . For
example for the function but the function does not have a point of inflection at . In fact, it is strictly concave up everywhere.
(ii)
In general, for a function , even if it has a point of inflection at and exists, it need not imply that
. For example, for the function has a point of inflection at .
11.2.15
Example:
Let us find the intervals of increase/ decrease, concavity upward/ downward, local maxima/ minima, and points of inflection (if any) of the function.
is as follows: 
represent the distance covered by a moving body up to time 
. Then, 
represents the velocity of the body at time 
and 
represents the acceleration of the body at time 
. Then, a point 
is a point of inflection means that if the body was accelerating till 
then it starts decelerating from the time 
onwards, or vice-versa. 
need not imply that the function 
has a point of inflection at 
. For 
but the function does not have a point of inflection at 
. In fact, it is strictly concave up everywhere. 
, even if it has a point of inflection at 
and 
exists, it need not imply that 
. For example, for the function 
has a point of inflection at 
. 
