Key Idea: The sign of the derivative tells you which way a curve is heading. Questions ask where is f increasing / decreasing? or read this graph of f′ — pure non-calculator work on Paper 1.
📈 The sign of f′ decides everything
- the gradient (derivative) at that x
- stationary — the curve is momentarily flat
To test a point, find f′ there and check its sign — not its size. f′(2) = −7 and f′(2) = −0.1 both just mean decreasing at x = 2.
🧭 Finding the intervals
👀 Reading a graph of f′
✏️ IB-style worked examples
IB-style question — increasing or decreasing at a point?
For f(x) = x² − 6x, the gradient function is f′(x) = 2x − 6. State whether f is increasing or decreasing at x = 1 and at x = 4.
Step by step:
Evaluate f′ at each point.
Read the signs.
Decreasing at x = 1; increasing at x = 4.
IB-style question — find where a function is increasing
Find the values of x for which f(x) = x² − 8x + 3 is increasing.
Step by step:
Differentiate, then set f′(x) > 0.
Solve the inequality.
Increasing for x > 4 (and decreasing for x < 4); the boundary x = 4 is the vertex.
IB-style question — intervals from a cubic
For f(x) = x³ − 12x, the gradient function is f′(x) = 3x² − 12. Find where f is increasing and where it is decreasing.
Step by step:
Stationary points: solve f′(x) = 0.
Test the sign of f′ in each region.
Increasing for x < −2 and x > 2; decreasing for −2 < x < 2.
IB-style question — read the graph of f′
The graph of f′ crosses the x-axis at x = 3, going from positive to negative. Explain why f has a local maximum at x = 3.
Step by step:
Left of 3, f′ > 0 → f increasing; right of 3, f′ < 0 → f decreasing.
Increasing then decreasing makes a peak.
f changes from increasing to decreasing at x = 3 (f′ goes + → −), so there is a local maximum there.
Important: Solving f′(x) = 0 only gives the boundaries, not the answer. You must test the sign of f′ between them — a cubic's f′ can be + − +, so f is increasing, then decreasing, then increasing again. Don't assume one side is increasing just because the other is.
Tap each card to reveal the answer.
Exam Tips
- Increasing where f′(x) > 0; decreasing where f′(x) < 0; stationary where f′(x) = 0.
- To find intervals: differentiate, solve f′(x) = 0 for the boundaries, then test the sign of f′ in each region.
- Only the sign of f′ matters — never its size.
- Reading a graph of f′: above the axis = increasing, below = decreasing.
- A + → − crossing of f′ is a maximum of f; a − → + crossing is a minimum.