The core rule: • f′(x) > 0 at a point → the function is increasing there (rising from left to right) • f′(x) < 0 at a point → the function is decreasing there (falling from left to right) • f′(x) = 0 at a point → the function is stationary there (flat, neither rising nor falling)
This is not a new calculation — it is simply reading the sign of the derivative you already know how to find.
| Gradient f′(x) | What the curve does | Picture |
|---|---|---|
| Positive (e.g. 3) | Rising — going uphill | ↗ |
| Zero | Flat — peak or valley | → |
| Negative (e.g. −5) | Falling — going downhill | ↘ |
Common mistake: Don't confuse where a function is large with where it is increasing. A function can be at a high value but still be decreasing (e.g. a ball at 40 m but falling).
To find where f is increasing or decreasing over a range, you need to know where f′(x) = 0 (the crossover points), then test the sign of f′ in each region.
Interval notation: IB accepts either inequality notation (x < −1) or interval notation (−∞, −1). Use whichever you find clearer. Always include the direction — 'increasing' or 'decreasing'.
Worked example
Apply the key method from Increasing and Decreasing Functions in a typical IB-style question.
Step by step
- Write the relevant formula or rule first.
- Substitute values carefully and show each step.
- State the final answer with correct units/context.
Final answer
Clear method and context-based interpretation secure most marks.
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A sign diagram is a quick visual tool that shows the sign of f′(x) across the x-axis. It replaces writing several sentences and is expected in many IB solutions.
How to draw a sign diagram: 1. Draw a horizontal line (this represents the x-axis). 2. Mark the x-values where f′(x) = 0. 3. Write + or − in each region based on a test value. 4. Under the x-axis, write ↗ for + and ↘ for −.
This single diagram immediately shows: increasing, decreasing, increasing.
Sign diagrams at endpoints: If the domain is restricted (e.g. 0 ≤ x ≤ 4), only consider the sign within that domain. Ignore what happens outside the given interval.
Worked example
Apply the key method from Increasing and Decreasing Functions in a typical IB-style question.
Step by step
- Write the relevant formula or rule first.
- Substitute values carefully and show each step.
- State the final answer with correct units/context.
Final answer
Clear method and context-based interpretation secure most marks.
In IB exam questions, increasing/decreasing analysis often appears inside a real-world context. The language changes but the maths is identical.
| Context word | Mathematical meaning | Test |
|---|---|---|
| Revenue is growing | R′(t) > 0 | Check sign of R′ |
| Temperature is falling | T′(t) < 0 | Check sign of T′ |
| Population is stable | P′(t) = 0 | f′ = 0 |
| Speed is increasing | v′(t) > 0 | Check sign of v′ |
IB exam language: If asked 'when is profit increasing?', answer with an interval AND a direction word: 'Profit is increasing for 0 < t < 4.' Just writing '0 < t < 4' with no context word may lose a mark.