In this section we consider some properties of curves which can be established using derivatives. These include intervals in which curves are increasing and decreasing, and the stationary points of functions.
The concepts of increasing and decreasing are closely linked to intervals of a function’s domain. Some examples of intervals and their graphical representations are:

For example: y = x 2 is decreasing for x

0 and increasing for x

0.
Sign diagrams for the derivative are extremely useful for determining intervals where a function is increasing or decreasing. Consider this example:

Remember that f(x) must be defined for all x on an interval before we can classify the interval as increasing or decreasing. We must exclude points where a function is undefined, and need to take care with vertical asymptotes.

EXERCISE 18D.1 Page 524
2. Find intervals where f(x) is increasing or decreasing:

STATIONARY POINTS
A stationary point of a function is a point such that f `(x) = 0.
It could be a local maximum, local minimum, or horizontal inflection.
HORIZONTAL OR STATIONARY POINTS OF INFLECTION
It is not always true that whenever we find a value of x where f0(x) = 0 we have a local maximum or mínimum

EXERCISE 18D.2 Page 527
Remember