Increasing/Decreasing Test and Critical Numbers
Increasing/Decreasing Test
 If $f '(x) > 0$ on an open interval, then $f$ is increasing on the interval.

If $f '(x) < 0$ on an open interval, then $f$ is decreasing on the interval.

(Note: when we ask "where is a function increasing?" we mean
"for which $x$values is a function increasing?")
We use this test in several ways. In order to determine whether a
function is increasing at a point $x=a$, you only need to see if
$f'(a)$ is positive. If you wish to know all places where a function
increases and decreases, you must find the sign of the derivative for
any values of $x$. Finding the intervals where $f'(x)$ is positive
(or negative), and hence where $f(x)$ is increasing (or decreasing)
is closely related to critical numbers.
Critical Numbers
Recall that a critical number (also called a critical point)
is a value of $x$ where
$f'(x)$ is either zero or doesn't exist.
We have already seen that
critical points are the only points (aside from endpoints) that can be
local maxima or minima. They have a second property that is almost
as important:
$f'(x)$ can only change sign at a critical point
The reason is simple. If $f'(x)$ changes sign continuously, then it has
to pass through 0 on its way from negative to positive (or viceversa).
That's the Intermediate Value Theorem. If $f'(x)$ changes sign discontinuously,
then there has to be a point where $f'(x)$ doesn't exist. Either way,
we have a critical point.
We will use critical numbers to find the intervals
where $f(x)$ is increasing and decreasing.
