Obviously, if the derivative of a function is differentiable, we can calculate its derivative, the derivative of the derivative, , which describes how the first derivative changes. And further we can calculate the third, fourth, etc. derivatives. Generally -th derivative will be denoted by and .
E.g., for , we get:
and for any ,
Condition for the existence of a local extremum
The following theorem holds. If for some :
- is differentiable in up to at least -th derivative,
- (respectively: ),
then has a minimum (respectively, maximum) in .
E.g. has maximum in , because:
Convex and concave functions, inflection points
We shall say that is convex on an interval (a,b), if its graph between any two points is under its secant line given by those points. In other words if for any and :
If the reverse inequality holds, the function is concave. If is convex on and concave on (or reversely) and continuous in , we will say that is an inflection point.
The following theorems hold:
- if exists on a given interval and is increasing (respectively, decreasing), then is convex (respectively, concave) on this interval,
- if exists on a given interval and is always positive (respectively, negative), then is convex (respectively, concave) on this interval,
E.g.: . We get , is positive on so is convex on this interval and negative on — so is concave on it. Point is an inflection point.