The diagram shows the curve y = xe^(-x²/4), for x ≥ 0, and its maximum point M. [Figure 9]

Mark scheme

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

1. [M1] (a) Use the correct product rule Condone error in chain rule.
2. [A1] (a) Obtain correct derivative in any form e.g. dy/dx = e^(-x²/4) - (x²/2) e^(-x²/4).
3. [DM1] (a) Equate derivative to zero and solve for x
4. [A1] (a) Obtain answer (√2, √2 e^(-1/2)) Or exact equivalent. Can state the components separately.
5. [B1] (b) State or imply dx = -(1/2)u^(-1/2) du Or equivalent e.g. du=2xdx. Alternative substitution: u = -x²/4.
6. [M1] (b) Substitute for x and dx
7. [A1] (b) Obtain correct integral ∫(1/2)e^(-u/4) du OE
8. [M1] (b) Use correct limits in an integral of the form ae^(-u/4) or ae^(-u/4) u=9 and u=0 or x=3 and x=0.
9. [A1] (b) Obtain answer 2 - 2e^(-9/4) Or exact equivalent.
10. [M1] (b) Alternative Method: ∫ xe^(-x²/4) dx = ae^(-x²/4) Recognition used.
11. [A1] (b) Alternative Method: a negative
12. [A1] (b) Alternative Method: a=-2
13. [M1] (b) Alternative Method: Use correct limits in an integral of the form ae^(-x²/4) x=3 and x=0.
14. [A1] (b) Alternative Method: Obtain answer 2 - 2e^(-9/4) Or exact equivalent.