A curve has equation x² – 6xy + 25y² = 16. Show that dy/dx = 0 at the point (3, 1). By finding the value of d²y/dx² at the point (3, 1), determine the nature of this turning point.

Mark scheme

Marking points from the official mark scheme (May/June 2015). The mark or mark type is shown in brackets.

1. [M1A1] 2x - 6(x dy/dx + y) + 50y dy/dx = 0
2. [A1A1] At (3,1) 6 - 18y' - 6 + 50y' = 0 ⇒ y' = 0 (AG)
3. [B1B1] 2 – 6(xy'' + y' + y') + 50[(y')² + yy''] = 0
4. [M1A1] At (3,1) 2 - 18y'' + 50y'' = 0 ⇒ y'' = -1/16
5. [A1] ⇒ maximum. (All previous marks required for final mark, i.e. CSO)