Topic 5.3 Notes – Determining Intervals on Which a Function Is Increasing or Decreasing

How the First Derivative Determines Increasing and Decreasing

Remember what the derivative means.
$f'(x)$ is the instantaneous rate of change of $f$. Geometrically, it’s the slope of the tangent line.

That slope tells you everything about behavior:

• If $f'(x) > 0$ → slope is positive → $f$ is increasing
• If $f'(x) < 0$ → slope is negative → $f$ is decreasing

Why this works:

• Positive slope means as $x$ increases, $f(x)$ increases.
• Negative slope means as $x$ increases, $f(x)$ decreases.

Here’s the key shift in thinking:
You are not plugging values into $f$ to see if outputs go up or down. You are analyzing the sign of $f'$.

On written responses, you must explicitly reference the derivative. Saying “the graph goes up” will not earn full credit. You need something like:

Since $f'(x) > 0$ on (1, 4), $f$ is increasing on (1, 4).

Where a Function Can Change Behavior

A function can only switch from increasing to decreasing (or vice versa) at specific x-values.

Critical Numbers

A critical number is any value of $x$ in the domain of $f$ where:

• $f'(x) = 0$, or
• $f'(x)$ is undefined

These are the only places where increasing/decreasing behavior can change.

Why? Because if $f'$ doesn’t hit zero or break, its sign can’t flip.

Points Where the Function Is Undefined

If $f(x)$ itself is undefined at some value, that also splits the domain into separate intervals.

Even if $f'(x)$ never equals zero, you must break the number line at those domain restrictions.

These values divide the number line into intervals where you test the sign of $f'$.

The Process for Finding Increasing and Decreasing Intervals

Let’s walk through the logic in order.

1. Find $f'(x)$.
2. Solve $f'(x) = 0$ to find critical numbers.
3. Find where $f'(x)$ is undefined.
4. Include any domain restrictions of $f$.
5. Use these values to divide the number line into intervals.
6. Pick a test point in each interval.
7. Plug into $f'(x)$ (not $f(x)$).
8. Determine the sign and state intervals in interval notation.

Quick Example

Suppose
$$f'(x) = (x-2)(x+1)$$

Critical numbers: $x = -1$ and $x = 2$

These split the number line into:

• $(-\infty, -1)$
• $(-1, 2)$
• $(2, \infty)$

Sign check:

So:

• Increasing on $(-\infty,-1)$ \cup $(2,\infty)$
• Decreasing on $(-1,2)$

Factoring makes sign analysis much faster than plugging random numbers.

Interpreting a Graph of $f'(x)$

Here’s a typical example of a derivative graph you might see on the exam.

Graph of $f'(x)$ with sign changes at $x=-2$ and $x=1$

How to read this:

• Where $f'(x)$ is above the x-axis, $f$ is increasing.
• Where $f'(x)$ is below the x-axis, $f$ is decreasing.
• Where $f'(x)=0$, behavior might change.

On the AP exam, they love giving a graph of $f'$ and asking about $f$. Don’t overthink it. Just look at whether the derivative is positive or negative.

Common Mistakes That Cost Points

• Assuming $f'(c)=0$ means increasing or decreasing at that point.
  A zero derivative only tells you slope is flat. You must check intervals around it.
• Testing $f(x)$ instead of $f'(x)$.
  Behavior comes from the derivative’s sign.
• Forgetting domain breaks.
  If $f$ is undefined at 3, you cannot include 3 in an interval.
• Not justifying with the derivative.
  Always connect the sign of $f'$ to the conclusion about $f$.