IB Mathematics: Applications and Interpretation SL — Topic 5 Calculus
Increasing and Decreasing Functions
Read the sign of the derivative to identify where a function rises, falls or is stationary.
Learning goal
Use the sign of the first derivative to identify increasing and decreasing intervals.
Syllabus link
AI SL: SL 5.2 · Topic 5 Calculus.
Big idea
The derivative converts graph behaviour into a sign question: positive means increasing; negative means decreasing.
Key relationship
\(f\'(x)>0\Rightarrow f\text{ increasing}\)
Core idea
The derivative converts graph behaviour into a sign question: positive means increasing; negative means decreasing.
Use the sign of the first derivative to identify increasing and decreasing intervals.
Key results and method
- Find or inspect \(f'(x)\).
- Locate the values where \(f'(x)=0\) or is undefined.
- Test the sign of \(f'\) on each interval and state the intervals clearly.
Worked example
For \(f(x)=x^3-3x\), find the intervals where \(f\) is increasing.
- \(f\'(x)=3x^2-3=3(x-1)(x+1)\).
- The derivative is positive for \(x<-1\) and \(x>1\).
Practice
Try these questions before moving on.
- Sketch a function whose derivative is positive, then negative, then positive.
- Use a derivative graph to state increasing intervals.
- Explain why \(f'(a)=0\) does not by itself prove a maximum or minimum.
Tutorials and premium resources
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