A function is increasing where and decreasing where . Find critical values where or is undefined, then use them to divide the domain into intervals and determine the sign of the derivative on each interval.
A function is increasing where and decreasing where . Find critical values where or is undefined, then use them to divide the domain into intervals and determine the sign of the derivative on each interval.
This is SL 5.2: Increasing and decreasing functions, linked to SL 5.6 on stationary points. A graphic display calculator (GDC) is permitted on every Mathematics: Applications and Interpretation paper.
The derivative (f'(x)) gives the gradient of the tangent to . Its sign therefore determines the function's local direction:
| Sign of (f'(x)) | Behaviour of |
|---|---|
| Positive gradient, so is increasing | |
| Negative gradient, so is decreasing | |
| Horizontal tangent; investigate both sides |
For example, let . Differentiate:
Solving gives the stationary x-values and . Test one value in each interval, such as , or inspect the factors:
| Interval | Sign of (f'(x)) | Behaviour |
|---|---|---|
| Positive | Increasing | |
| Negative | Decreasing | |
| Positive | Increasing |
Therefore, is increasing on and , and decreasing on . At , the sign changes from positive to negative, giving a local maximum. At , it changes from negative to positive, giving a local minimum.
A reliable procedure is: first differentiate; second solve and note points where (f') is undefined; third place these values on a number line; fourth test one value from each interval. On a GDC, graph (f') or use a table to check signs, but write the resulting intervals in mathematical notation. Keep calculator values unrounded until the final answer. If the domain is restricted, compare only intervals within that domain.
A common misconception is that automatically means a maximum or minimum. It only identifies a stationary point. For example, has but no local extremum. Also, only the sign of (f'(x)) matters, not whether is positive or negative.
Show (f'(x)), solve , and provide a sign table or clear interval reasoning. Respect any restricted domain and check endpoints when asked for a greatest or least value.