Topic 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing: use the sign of the first derivative to find intervals of increase and decrease.

What this topic is asking

The College Board (Topic 5.3) connects the sign of $f'(x)$ to the behavior of $f$. Where $f'(x) > 0$ the function is increasing; where $f'(x) < 0$ it is decreasing. You must build a sign chart for $f'$ and translate it into intervals, with justification that names the sign of the derivative.

The principle

Building a sign chart

The reliable method is mechanical. Find $f'(x)$, factor it, mark its zeros and points of undefinedness on a number line, then pick a test value in each interval and record the sign of $f'$. Because $f'$ is continuous between consecutive critical points, one test value determines the sign across the whole interval.

Open versus closed intervals and justification

Report intervals of increase and decrease as open intervals; the function's behavior at an isolated critical point (where $f' = 0$) is a single instant, not an interval. On free-response questions the marks live in the justification: you must write that $f$ is increasing "because $f'(x) > 0$ on this interval", not merely state the interval. A common scoring rule is that the answer is worth nothing without the sign-of-derivative reason. When the derivative is given as a graph rather than a formula, read the sign directly from whether the graph of $f'$ is above or below the $x$-axis.

Why one test value settles a whole interval

The reason a single test point fixes the sign across an entire subinterval is that $f'$ can only change sign at its zeros or where it is undefined, the critical points. Between two consecutive critical points $f'$ is continuous and never zero, so by the Intermediate Value Theorem it cannot switch from positive to negative without passing through zero, which would be another critical point. It therefore keeps one sign throughout, and any convenient test value reveals that sign. This is why the sign-chart method is valid and efficient: you are not sampling at random but exploiting the fact that the critical points are the only places the direction can flip. Choosing easy test values, such as $0$ or a whole number well inside the interval, keeps the arithmetic light on the no-calculator section.

The connection to extrema

Intervals of increase and decrease set up the classification of critical points. Where a function switches from increasing to decreasing it reaches a local maximum, and where it switches from decreasing to increasing it reaches a local minimum, which is precisely the First Derivative Test of the next topic. So the sign chart you build here is reused immediately: the same positive-to-negative or negative-to-positive transitions that you read off to describe behavior also classify the turning points. Seeing increasing/decreasing analysis as the foundation for extrema, rather than a separate exercise, lets you answer both kinds of question from one sign chart, which the exam often asks you to do in adjacent parts of a single free-response problem.