TheIBAxolotl
The sign tells the story
If you walk along a curve from left to right, the derivative tells you whether you are going uphill or downhill:
- \(f'(x) > 0\): the function is increasing (uphill),
- \(f'(x) < 0\): the function is decreasing (downhill),
- \(f'(x) = 0\): the tangent is horizontal, a possible turning point.
So to find where a function increases, find where its derivative is positive. That turns a graph question into an inequality.
Worked example
Find where \(f(x) = x^3 - 3x^2 - 9x + 2\) is increasing and decreasing.
- Differentiate: \(f'(x) = 3x^2 - 6x - 9 = 3(x^2 - 2x - 3) = 3(x - 3)(x + 1)\).
- \(f'(x) = 0\) at \(x = -1\) and \(x = 3\).
- Test the sign in each region:
| Interval | \(x < -1\) | \(-1 < x < 3\) | \(x > 3\) |
|---|---|---|---|
| Test value | \(x = -2\) | \(x = 0\) | \(x = 4\) |
| \(f'(x)\) | \(15 > 0\) | \(-9 < 0\) | \(15 > 0\) |
| \(f\) is | increasing | decreasing | increasing |
So \(f\) increases for \(x < -1\) and \(x > 3\), and decreases for \(-1 < x < 3\). This also tells us there is a local maximum at \(x = -1\) and a local minimum at \(x = 3\).
Reading the graph of \(f'\)
Exams often give the graph of \(f'\) and ask about \(f\). Keep a translation in your head:
- where the \(f'\) graph is above the \(x\)-axis, \(f\) is increasing;
- where it is below, \(f\) is decreasing;
- where \(f'\) crosses the axis from positive to negative, \(f\) has a maximum; from negative to positive, a minimum;
- where \(f'\) touches the axis without changing sign, \(f\) has a horizontal point that is not a turning point.
A common trap is to look at whether the \(f'\) graph is rising. That tells you about the second derivative, not about whether \(f\) increases.
Reading the graph of \(f\)
Going the other way, from \(f\) to \(f'\): steep uphill sections of \(f\) become large positive values of \(f'\), flat sections give zeros, and downhill sections give negative values. Sketching \(f'\) from \(f\) is just recording slopes.
Graphical interpretation in context
If \(P(t)\) is a population, \(P'(t) > 0\) means the population is growing. A question asking "when is the population decreasing?" is a question about the sign of \(P'\).
Common mistakes
- Including endpoints like \(x = -1\) in a strictly increasing interval without care (both conventions can be accepted, but be consistent and follow the question).
- Judging the sign of \(f'\) from whether its graph is going up rather than whether it is above the axis.
- Solving \(f'(x) = 0\) and stopping without testing the signs around each root.
- Forgetting that the derivative's zeros only suggest turning points.
Exam tips
- A sign table is quick and shows clear method.
- Give intervals with inequalities in \(x\), not in \(y\).
- On Paper 2, graph \(f'\) on the GDC and read where it is above or below the axis.