The equation of a curve is y = cos³x√sinx. It is given that the curve has one stationary point in the interval 0 < x < ½π. Find the x-coordinate of this stationary point, giving your answer correct to 3 significant figures.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. M1 Use the correct product rule and then the chain rule to differentiate either cos³ x or √sin x e.g. two terms with one part of dy/dx = pcos²xsin x√sinx + qcos³ x cos x/√sinx
2. A1 Obtain correct derivative in any form e.g. dy/dx = -3 cos²x sinx√sinx + cos³ x cos x/(2√sinx) A1 for each correct term substituted in the complete derivative.
3. M1 Equate their derivative to zero and obtain a horizontal equation with positive integer powers of sin x and/or cos x from an equation including √sinx or 1/√sinx using sensible algebra. e.g. -3 cos²xsin²x + ½cos⁴x = 0
4. M1 Use correct formula(s) to express their equation/derivative in terms of one trigonometric function Can be awarded before the previous M1. May involve more than one trigonometric term.
5. A1 Obtain 7cos²x=6, 7sin²x=1, or 6tan²x=1, or equivalent, and obtain answer x = 0.388 CAO. The question asks for 3 sf. Ignore additional answers outside (0, ½π). 22.2° is A0.