Increasing and Decreasing Functions
Sign of f′ tells the direction.
Key idea
Sign of f′ tells the direction.
In depth
If f′ > 0 on an interval, f is increasing; if f′ < 0, decreasing. Sign changes of f′ occur at critical points, so test a point in each region between them.
Worked example
f(x) = x² − 4x: f′ = 2x − 4. Decreasing for x < 2, increasing for x > 2.
Real-world uses
- Predicting when a company’s profit will rise or fall
- When a river level is going up
Common mistake
Using f instead of f′ when testing the sign.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Increasing and Decreasing Functions?
Sign of f′ tells the direction. It is a topic at the advanced level in Calculus.
Where is Increasing and Decreasing Functions used in real life?
Real-world examples: Predicting when a company’s profit will rise or fall; When a river level is going up.
What should I learn before Increasing and Decreasing Functions?
Make sure you are comfortable with Critical Points, the previous topic in Calculus.
What is a common mistake with Increasing and Decreasing Functions?
Using f instead of f′ when testing the sign.
How can I practise Increasing and Decreasing Functions?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Increasing and Decreasing Functions?
Move on to First Derivative Test next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.