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 $f'$.

Sign analysis of $f'$ is the skill behind finding relative extrema, locating absolute extrema, and sketching the graph of $f$ from information about $f'$, so it comes up all through Unit 5.

Using the Sign of f' to Find Intervals

• $f'(x) > 0$ 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.

Justifying Increasing and Decreasing Intervals With f'

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

• Misconception: A high point on the graph of $f'$ means $f$ is large.
  Actually: 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.
• Misconception: Saying "it's going up" is enough to justify an answer.
  Actually: 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.
• Misconception: Points where $f$ is undefined do not need to be included.
  Actually: The function can only change direction at critical points and at points outside its domain, so include both when you split the number line.
• Misconception: A zero of $f'$ always means a max or min.
  Actually: 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.
• Misconception: Intervals of increase and decrease should be reported as closed intervals.
  Actually: Report intervals of increase and decrease as open intervals between the critical points.
• Misconception: It is acceptable to write "it" instead of naming the function.
  Actually: Always refer to $f$, $f'$, and $f''$ by name so your reasoning is unambiguous.

Related AP Calculus Guides

• Unit 5 Review: 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