The diagram shows the part of the curve y = x² cos 3x for 0 ≤ x ≤ π/6, and its maximum point M, where x = a. [Figure 5.1]

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

1. [M1] Use correct product rule; d/dx(x²)cos(3x) + x² d/dx(cos 3x).
2. [A1] Obtain correct derivative in any form; e.g. 2xcos3x-3x² sin 3x .
3. [A1] Equate derivative to zero and obtain a = (1/3) tan⁻¹(2/(3a)); AG Condone a = (1/3) tan⁻¹(2/(3a)) Must at least reach expression 2x = 3x² tan(3x) or better before final answer to gain A1. Final answer must be in terms of a. Can work with x and switch to a at very end. Look for (2/3)a or (2/3)x in working not immediately corrected or as penultimate line A0.
4. [M1] Use the iterative process a(n+1) = (1/3) tan⁻¹(2/(3a(n))) correctly at least twice during successive iterations in the numerous iterations; Degrees 0/3.
5. [A1] Obtain final answer 0.36; Must be 2d.p.
6. [A1] Show sufficient iterations to 4 or more d.p. to justify 0.36 to 2 d.p. or show there is a sign change in the interval (0.355, 0.365); Allow small errors in 4th d.p. Allow errors at start if self corrects later. Example iterations: 0.3091 -> 0.3435 -> 0.3826 -> 0.4264 -> 0.4740 -> 0.3017 -> 0.3989 -> 0.3789 -> 0.3650 -> 0.3499 -> 0.3339 -> 0.3176 -> 0.3820 -> 0.3439 -> 0.3513 -> 0.3566 -> 0.3625 -> 0.3688 -> 0.3754 -> 0.3502 -> 0.3649 -> 0.3619 -> 0.3599 -> 0.3576 -> 0.3552 -> 0.3526 -> 0.3624 -> 0.3567 -> 0.3578 -> 0.3604 -> 0.3614 -> 0.3576 -> 0.3580