The diagram shows the curve with equation y = 3x² ln((1/2)x). The curve crosses the x-axis at the point P and has a minimum point M. [Figure 8]

Mark scheme

Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.

1. [*M1] Apply product rule to find first derivative
2. [A1] Obtain 6xln((1/2)x)+3x or equivalent
3. [B1] Identify x = 6 at P
4. [DM1] Substitute their value of x at P into attempt at first derivative
5. [A1] Obtain 18
6. [*M1] Equate their first derivative to zero and attempt solution of equation of form kln((1/2)x)+m=0
7. [DM1] Obtain x-coordinate of form ae^(b)
8. [A1] Obtain x = 6e⁻¹ or exact equivalent
9. [M1] Substitute exact x-value in the form a₂e^(b) and attempt simplification to remove ln
10. [A1] Obtain -54e⁻¹ or exact equivalent