Key Idea: This topic uses the derivative to find the peaks, troughs and bends of a curve and to solve real max/min problems. It's almost all Paper 1 by-hand: differentiate, set the derivative to zero, solve, then justify the nature.
⛰️ Stationary points: find, classify, locate
A stationary point is where the gradient is zero. Find it by solving f′(x) = 0 (factor the derivative first), classify it with the table below, then get the y-coordinate by substituting x into the original f(x) — not into f′(x), which just gives 0.
🔧 Optimisation & inflexion
A point of inflexion is where the curve changes concavity. Solve f″(x) = 0 for candidate x-values, then confirm f″ changes sign across each. f″ = 0 alone is not enough — e.g. y = x⁴ has f″(0) = 0 but no inflexion.
✏️ IB-style worked examples
IB-style question — find and classify stationary points
Find and classify the stationary points of f(x) = x³ − 3x² − 9x + 2, giving their coordinates.
Step by step:
Differentiate and solve f′(x) = 0 (factor first).
Classify with the second derivative.
Substitute into the ORIGINAL f for each y-coordinate.
Maximum (−1, 7); minimum (3, −25).
IB-style question — maximise an area (optimisation)
A rectangular enclosure uses a wall as one side and 60 m of fencing for the other three sides. With width x, find the width that gives the maximum area, and that maximum area.
Step by step:
Model with the constraint (two widths + one length = 60).
Differentiate, solve A′(x) = 0, and justify the maximum.
Answer what's asked — the maximum area, not just x.
Width 15 m gives the maximum area of 450 m².
IB-style question — point of inflexion (confirm the sign change)
Find the point of inflexion of f(x) = x³ − 6x² + 4x + 1.
Step by step:
Differentiate twice and solve f″(x) = 0.
Confirm f″ changes sign across x = 2.
Get y from the original function.
Point of inflexion at (2, −7).
Important: Don't stop at f′(x) = 0. You must justify whether it's a max or min (f″, or the sign of f′), and then give what's actually asked — the maximum volume or the dimensions, with units, not just the value of x. For inflexion, always confirm f″ changes sign.
Tap each card to reveal the answer.
Exam Tips
- Stationary points: solve f′(x) = 0 (factor the derivative first).
- Classify with f″ (>0 min, <0 max); if f″ = 0, test the sign of f′ either side.
- y-coordinate always comes from the ORIGINAL function f, never f′.
- Optimisation: get to ONE variable via the constraint, then answer what's asked (with units).
- Inflexion needs f″(x) = 0 AND a sign change of f″ — check both sides every time.