SL 5.2—Increasing and decreasing functions
- Syllabus
- First assessment 2021
- Objective
- —
- Level
- HL
The sign of a derivative controls local increase and decrease
Where f'(x)>0, f is increasing locally; where f'(x)<0, f is decreasing locally. A stationary point has f'(x)=0, but the sign must be checked on either side to classify the change.
Find critical x-values, split the domain into intervals and test the derivative sign. This produces a variation table and keeps domain restrictions visible instead of relying on a sketch.
For f(x)=x³−3x, f'(x)=3x²−3. The derivative is positive outside x=−1 and x=1 and negative between them, so the function rises, falls, then rises again.
f'(x)=0 does not automatically mean a maximum or minimum. A horizontal inflection can have zero derivative without changing from increasing to decreasing.