Mathematics · May / June 2026 · Paper 43 · Question 20

[ d y x =] 24x2 – 6 d setting their derivative = 0 or d y x = 0 d 1 , 3 1 , 7 − and 2 2 [20]

B2 B1 for one correct term, 24x2 or – 6, seen For B2 or B1, condone +0. For B2 or B1, isw 2nd derivative. For B2, condone poor notation, e.g. y = 24x2 – 6. M1 dep on at least B1 earned M1 can be implied from a correct method to find x or from correct solutions but must use their first derivative. After B2, must use correct derivative = 0 to score M1.

B2 B1 for x = 1 1 − 2 and x = 2 or for 1 , 3 1 , 7 − or 2 2

1 , 7 − maximum with 2 correct reason AND 1 , 3 minimum with 2 correct reason [3]

Reasons could be e.g. 1 , 7 and 1 , 3 − For 3 marks or B2, allow to 1 A reasonable sketch of a positive cubic 2 2 2 Correct use of 2nd derivative: be implied from working. 1 , 7 48 × –0.5 < 0 or –24 < 0, so − For 3 marks or B2, accept reasonable sketch drawn is max oe. 2 in part (b) or referred to if drawn in part (a), accept without axes. 48 × 0.5 > 0 or 24 > 0, so 1 , 3 is a min oe. 2 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. For B3 or B2, if drawn on axes, minimum must be in the 1st quadrant and the maximum in the second

B2 for 1 correct with correct reason for that quadrant. stationary point or for both x-values correct and reasonable y values may be incorrect sketch of a positive cubic, or for correct substitution and evaluation of both The second derivative must be correct FT from of their x-values into their second derivative their first derivative for B2 or M1, accept poor notation or substitution and evaluation for one x-value on May use substitution into their dy/dx with stated both sides of both of their stationary points to x-values find the gradients soi or the values of y

or M1 for showing [2nd derivative =] 48x Accept poor notation. or correct FT their 1st derivative or substitution and evaluation shown for one x- May use substitution into their dy/dx value on both sides of one of their stationary points to find the gradients soi or the values of y or for sketch of any positive cubic with incorrect answers.