A function f is such that f'(x) = 6(2x−3)² − 6x for x ∈ R.

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

1. [B1*] 6(2x-3)²-6x<0 or = 0 Condone < 0. If 6(2x-3)² only or 6(2x-3)² -6 is used, do not treat as a MR.
2. [M1] 24x² -78x+54 or 4x² -13x+9 or (x−1)(4x-9) OR 6(2x-3)² <6x leading to (2x-3) <√x leading to 2x - √x - 3 Expanding brackets and collecting terms to arrive at a three term quadratic, only condone sign errors.
3. [B1] [x=]1, 9/4
4. [DB1FT] 1<x<9/4 or x>1 and x < 9/4 or (1, 9/4) OE Condone consistent use of < and ≥ or [ ]. Do not allow x >1or x < 9/4 nor x >1,x < -9/4. FT on their values coming from a correct initial statement.
5. [B1] [f(x)=] 6(2x-3)³ / (3*2) - 6x²/2 [+C] B1 for each {Correct integral} .
6. [M1] -1=(-1)³-3x1² + C f(x) = -1 equated to their integrated expression, defined by two terms with at least one correct power + C, with x = 1.
7. [A1] [f(x)=](2x-3)³-3x² + 3 CAO Only condone C = 3 as final answer if coefficients have been simplified earlier.