Increasing/Decreasing Test and Critical Numbers
How can we tell if a function is increasing or decreasing?
When we discuss where a
function is increasing or decreasing we mean for which
$x$values is a function increasing or
decreasing.
If we have a function of time, we might discuss when a function is increasing or
decreasing, and we are talking about for which $t$values is a function
increasing or decreasing.
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.

DO: Ponder the graphs
in the box above until you are confident of why the two
conditions listed are true.
You should completely master this concept; a helpful shorthand:
$$f'>0\Longleftrightarrow f
\uparrow$$ $$ f'<0 \Longleftrightarrow f\downarrow $$
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 $x$.
DO: What do we know
about whether $f$ is increasing or decreasing at $x=a$ if
$f'(a)=0$?
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 $f$ is defined, and where $f'(x)$ is
either zero or doesn't exist. We have already seen that critical numbers are the only $x$values that
can be local maxima or minima. They have a
second property that is almost as important:
$f'$ can only change sign at a
critical number.
The reason is simple. If $f'(x)$ is continuous and it changes
sign, then it has to pass through 0 on its way from negative to
positive (or vice versa). That's the Intermediate
Value Theorem. If $f'(x)$ is not continuous where it changes
sign, then that is a point where $f'(x)$ doesn't exist. Either
way, $f'$ changes sign only at a critical number.
Thus, $f$ can change
direction only at a critical number.
We will use critical numbers to find the intervals where $f(x)$
is increasing and decreasing.
