AP® Calculus BC review sheet from Aim for Five (aimforfive.com/calc-bc/units/5/5-3)
Unit 5 · Topic 5.3
5.3 Determining Intervals on Which a Function Is Increasing or Decreasing
The sign of the derivative tells you the direction of the function. Where f′ > 0, f is increasing; where f′ < 0, f is decreasing. To find these intervals, locate the critical points and test the sign of f′ between them.
Key terms
- increasing
- decreasing
- sign chart
- critical point
- first derivative
The connection
If f′(x) > 0 for every x in an interval, f is increasing on that interval: moving right, the graph goes up. If f′(x) < 0 on an interval, f is decreasing there.
This follows from the Mean Value Theorem: between any two points in the interval, the average rate of change equals f′(c) for some c, and if every f′(c) is positive, every average rate is positive too.
The sign chart method
Follow these steps:
- Find f′(x) and factor it as much as possible.
- Find the critical points (f′ = 0 or undefined) and any points where f is undefined. Mark them on a number line.
- Pick a test value in each interval and find the sign of f′ there. Or reason from the sign of each factor.
- Read off the intervals: f′ > 0 means increasing, f′ < 0 means decreasing.
Signs from factors
If f′ is factored, you often don't need to compute anything. For f′(x) = 3(x − 3)(x + 1), the factor (x − 3) is negative for x < 3, and (x + 1) is negative for x < −1. Multiply the signs on each interval. Exponential factors like e⁻ˣ are always positive, so they never change the sign. Squared factors like (x − 2)² are never negative, so they don't cause a sign change.
Writing the justification
A complete answer gives the interval and the reason: “f is increasing on (−∞, −1) and (3, ∞) because f′(x) > 0 there.” A sign chart by itself usually isn't enough for full credit on free response; write the sentence too.
If you're given a graph of f′ instead of a formula, read the signs directly: f is increasing where the graph of f′ is above the x-axis, and decreasing where it is below. Don't confuse this with where f′ itself is increasing.
Open or closed intervals?
If f is continuous at an endpoint of an interval where it's increasing, you may include that endpoint: for f(x) = x², both “increasing on (0, ∞)” and “increasing on [0, ∞)” are accepted. Never include a point where f is undefined.
Don't join separate intervals with a union sign unless f really increases across the gap. For f(x) = −1/x, f′(x) = 1/x² > 0 everywhere it's defined, but f is not increasing on (−∞, 0) ∪ (0, ∞) as a whole: f(−1) = 1 is bigger than f(1) = −1. Say “increasing on (−∞, 0) and on (0, ∞).”
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Intervals from a polynomial
Find the intervals where f(x) = x³ − 3x² − 9x + 4 is increasing and decreasing.
Show the solutionHide the solution
- Step 1: f′(x) = 3x² − 6x − 9 = 3(x − 3)(x + 1). Critical points: x = −1 and x = 3.
- Step 2: x < −1: test x = −2: 3(−5)(−1) = 15 > 0.
- Step 3: −1 < x < 3: test x = 0: 3(−3)(1) = −9 < 0.
- Step 4: x > 3: test x = 4: 3(1)(5) = 15 > 0.
Answer: f is increasing on (−∞, −1) and (3, ∞), and decreasing on (−1, 3), because f′ > 0 and f′ < 0 on those intervals respectively.
- Example 2
An exponential factor
Where is f(x) = xe⁻ˣ increasing?
Show the solutionHide the solution
- Step 1: Product rule: f′(x) = e⁻ˣ + x(−e⁻ˣ) = e⁻ˣ(1 − x).
- Step 2: e⁻ˣ > 0 always, so the sign of f′ matches the sign of (1 − x).
- Step 3: 1 − x > 0 when x < 1.
Answer: f is increasing on (−∞, 1) because f′(x) = e⁻ˣ(1 − x) > 0 there (and decreasing on (1, ∞)).
- Example 3
Trap: a critical point with no sign change
f′(x) = (x − 2)²(x + 1). On what intervals is f increasing?
Show the solutionHide the solution
- Step 1: Critical points: x = −1 and x = 2.
- Step 2: (x − 2)² ≥ 0 everywhere, so the sign of f′ comes from (x + 1).
- Step 3: x < −1: f′ < 0. −1 < x < 2: f′ > 0. x > 2: f′ > 0.
- Step 4: f′ doesn't change sign at x = 2. It's 0 at that single point, but positive on both sides.
Answer: f is increasing on (−1, 2) and (2, ∞), and in fact on all of (−1, ∞), because f′ > 0 there except at the single point x = 2. It is decreasing on (−∞, −1).
Common mistakes
- Assuming the sign alternates at every critical point. Squared factors don't change sign.
- Reading where f′ is increasing (its slope) instead of where f′ is positive (its sign) on a graph of f′.
- Giving intervals in terms of y-values instead of x-values.
On the exam
- Free-response questions frequently give a graph of f′ and ask on which intervals f is increasing. Answer with x-intervals and the reason “because f′ > 0.”
- Watch for points where f is undefined; intervals of increase can't cross them.
Connected topics
- Unit 55.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
- Unit 55.4 Using the First Derivative Test to Determine Relative (Local) Extrema
- Unit 55.9 Connecting a Function, Its First Derivative, and Its Second Derivative
- Unit 44.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
Videos
Check yourself
4 questions on 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing. Pick an answer to see if you got it, and why.
Let f(x) = x⁴ − 4x³. What is the largest open interval on which f is decreasing?
The derivative of a function f is given by f′(x) = x sin x − 1 for 0 < x < 5. On which of the following intervals is f increasing?
On which of the following intervals is f(x) = x²e^(−x) increasing?
Let f(x) = ln(x² + 1) − x. On which of the following intervals is f increasing?
0 of 4 answered