5.3 Determining Intervals on Which a Function is Increasing or Decreasing

A function is increasing where its first derivative is positive and decreasing where the first derivative is negative. To find these intervals, locate the critical points (where $f'(x)=0$ or $f'$ is undefined) plus any points where the function itself is undefined, split the domain at those points, then test the sign of $f'$ in each piece. For AP Calculus, justify each interval with the sign of $f'$.

Why This Matters for the AP Calculus Exam

This topic is the backbone of analyzing function behavior with derivatives, which shows up across Unit 5. On the AP Calculus exam you will use sign analysis of $f'$ to justify where a function increases or decreases, and that same skill leads directly into finding relative extrema, locating absolute extrema, and sketching graphs of $f$ from information about $f'$.

These questions appear in both multiple-choice and free-response settings. Strong justifications matter here: when you claim a function is increasing, you need to point to the sign of the derivative, not just say "the graph goes up." Refer to $f$, $f'$, and $f''$ by name so your reasoning is clear.

Key Takeaways

• $f'(x) > 0$ means $f$ is increasing; $f'(x) < 0$ means $f$ is decreasing.
• A function can only switch between increasing and decreasing at critical points (where $f'(x)=0$ or $f'$ is undefined) or where $f$ itself is undefined.
• Split the domain at those points, then test the sign of $f'$ at one sample point inside each interval.
• Report your answer in interval notation using open intervals.
• Justify with the sign of $f'$, not vague phrases like "it's going up."
• Always name $f$, $f'$, or $f''$ so the reader knows which function you mean.

When Does a Function Increase or Decrease?

The derivative of a function gives its rate of change at a point, so the sign of the derivative tells you which direction the function is heading.

1. If $f'$ is positive at a point, $f$ is increasing there.
2. If $f'$ is negative at a point, $f$ is decreasing there.

The graph below shows this pattern. The gray line is the function $f$, and the black line is its derivative $f'$.

A function can only change direction (from increasing to decreasing or the reverse) at its critical points, where the derivative equals $0$ or is undefined, and at points where the function itself is undefined.

So for each interval between those points, pick one sample $x$-value, plug it into $f'$, and check the sign. Positive means increasing on that interval; negative means decreasing. The sign stays the same across an interval because the function cannot switch direction without passing through a critical point.

Worked Example

Let $h$ be a function defined for all real numbers except $0$, with derivative $h'(x)=\frac{(x+7)}{x^2}$. On which intervals is $h$ increasing?

Find where $h$ can change direction. Set $h'(x)=0$:

$h'(x)=\frac{(x+7)}{x^2}$

$0=\frac{(x+7)}{x^2}$

This gives $h'=0$ at $x=-7$, so that is a critical point. The function is also undefined at $x=0$, as stated. These two points split the number line into three intervals:

$(-\infty,-7),(-7,0),(0,\infty)$

Test a sample $x$-value in each interval. Here we use $x=-8$, $x=-1$, and $x=1$.

So $h$ is increasing on $(-7,0)$ and $(0,\infty)$.

Steps to Determine Function Behavior

1. Find the critical points, where $f'(x)$ equals zero or is undefined.
2. Split the domain into intervals at those critical points (and any points where $f$ is undefined).
3. Choose a test point inside each interval.
4. Evaluate $f'$ at each test point.
5. Read the sign: positive means $f$ is increasing on that interval, negative means $f$ is decreasing.

How to Use This on the AP Calculus Exam

Problem Solving

Use the sign chart of $f'$ as your main tool. Find critical points, split the domain, test each interval, and translate the sign of $f'$ into increasing or decreasing behavior of $f$.

Free Response

When a question says "Justify your answer," anchor your justification to the sign of the derivative. Write something like "$f$ is increasing on $(-7,0)$ because $f'(x)>0$ on that interval." Naming $f$ and $f'$ specifically keeps your reasoning clear and easy to follow, which is important for clean exam work.

Common Trap

If you are given the graph of $f'$ instead of an equation, read where the graph is above the $x$-axis (positive, so $f$ is increasing) and below it (negative, so $f$ is decreasing). Do not treat the height of the $f'$ graph as the height of $f$.

Practice Problems

Question 1

Let $f(x)=x^3-27x$. On which interval(s) is $f$ decreasing?

Question 2

Let $f(x)=x^{4}-2x^{2}$. On which interval(s) is $f$ increasing?

Answers and Solutions

Question 1

The answer is $(-3,3)$.

Look for where $f'$ is negative. The derivative is $f'(x)=3x^2-27$, which equals $0$ at $x=-3$ and $x=3$, so these are the only points where $f$ can change direction. They split the number line into three intervals: $(-\infty,-3),(-3,3),(3,\infty)$.

So $f$ is decreasing on $(-3,3)$.

Question 2

The answer is $(-1,0)$ and $(1,\infty)$.

Look for where $f'$ is positive. The derivative is $f'(x)=4x^3-4x$, which factors as $4x(x+1)(x-1)$. It equals $0$ at $x=-1$, $x=0$, and $x=1$, so these are the only points where $f$ can change direction. They split the number line into four intervals: $(-\infty,-1),(-1,0),(0,1),(1,\infty)$.

So $f$ is increasing on $(-1,0)$ and $(1,\infty)$.

Common Misconceptions

• Reading the $f'$ graph as if it were $f$. A high point on the graph of $f'$ does not mean $f$ is large; it means $f$ is increasing quickly. Use the sign of $f'$, not its height.
• Justifying with "it's going up." On free-response questions you need to cite the sign of the derivative, such as "$f'(x)>0$ on this interval," not a casual description.
• Forgetting points where $f$ is undefined. The function can only change direction at critical points and at points outside its domain, so include both when you split the number line.
• Saying $f'=0$ always means a max or min. A zero of $f'$ marks a possible change in direction, but $f$ might keep increasing or decreasing through it if the sign does not switch.
• Using closed intervals carelessly. Report intervals of increase and decrease as open intervals between the critical points.
• Writing "it" instead of naming the function. Always refer to $f$, $f'$, and $f''$ by name so your reasoning is unambiguous.

Related AP Calculus Guides

• Unit 5 Overview: Analytical Applications of Differentiation
• 5.1 Using the Mean Value Theorem
• 5.2 Extreme Value Theorem, Global vs Local Extrema, and Critical Points
• 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
• 5.11 Solving Optimization Problems
• 5.10 Introduction to Optimization Problems