The diagram shows part of the curve y = 15/x - 5/x². The curve meets the x-axis at the point A. The curve has a maximum at the point B. Find the area of the shaded region enclosed by the line AB and the curve. Give your answer in exact form.

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [B1] For A: x=1/3
2. [B2] dy/dx = -15/x² + 10/x³ oe. isw (B1 for each correct term)
3. [M1] Solves their dy/dx = 0 as far as x = ...
4. [A2] B: (2/3, 45/4) oe (A1 for each correct coordinate)
5. [M1] Correct plan e.g. ∫(their 15/x - 5/x²)dx - 1/2 x their 45 x (their 2/3 - their 1/3)
6. [B2] ∫(15/x - 5/x²)dx = 15lnx + 5/x oe (B1 for 15lnx or klnx + 5/x oe)
7. [M1] Substitute limits for integration and subtract area of triangle/trapezium, e.g. (15ln(2/3) + 15/2) - (15ln(1/3) + 15) for integral, and 1/2 x (1/3) x 45/4 for triangle
8. [A1] Area of shaded region: 15ln2 – 75/8 or exact equivalent