Mathematics · Oct / Nov 2024 · Paper 42 · Question 10

3x2 – 18x – 48 final answer [2]

B1 for two correct terms or for correct answer seen then spoiled

their d y = 0 soi d x [3](x – 8)(x + 2) oe −− ± − − − ( 18) ( 18) 2 4(3)( 48) or × 2 3 oe or + 2 3 ± 16 3 (–2, 52) (8, –448) [5]

M1 M1 ± oe 18 900 6

B2 B1 for one correct pair of coordinates or for two correct values of x

(–2, 52) maximum with reason and ( ) − 8, 448 minimum with reason and no incorrect statement [3]

Reasons could be e.g. 1. A reasonable sketch of a positive cubic 2. Correct evaluation and use of 2nd derivative 6x – 18 = –30, –30 < 0, so (–2, 52) is a maximum oe. 6x – 18 = 30, 30 > 0 , so ( ) − 8, 448 is a minimum oe. 3. Evaluates correctly values of y on both sides of both correct stationary points 4. Finds gradient on each side of both correct stationary points. Any incorrect statement MAX

B2 B2 for 1 correct with correct reason for that stationary point or for both x-values correct and reasonable sketch of a positive cubic or for correct substitution and evaluation of both of their x-values into their second derivative or substitution and evaluation for one x-value on both sides of both of their stationary points to find the gradients soi

or M1 for showing [2nd derivative =] 6x – 18 or correct FT their 2nd derivative from part (a) or substitution and evaluation shown for one x- value on both sides of one of their stationary points to find the gradients soi or for sketch of any positive cubic.