October 2021 Paper 2 Q5

Notes

(a)(i) M1: \(x^n \rightarrow x^{n-1}\) for at least one power of \(x\)

A1: \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 20x^3 - 72x^2 + 84x - 32\)

(a)(ii) A1ft: Achieves a correct \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\) for their \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 20x^3 - 72x^2 + 84x - 32\)

Notes

(b)(i) M1: Substitutes \(x = 1\) into their \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\)

A1: Obtains \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0\) following a correct derivative and makes a conclusion which can be minimal e.g. tick, QED etc. which may be in a preamble e.g. stationary point when \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0\) and then shows \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0\)

Alternative for (b)(i)

M1: Attempts to solve \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0\) by factorisation. This may be by using the factor of \((x - 1)\) or possibly using a calculator to find the roots and showing the factorisation. Note that they may divide by 4 before factorising which is acceptable. Need to either see either \(4(x - 1)^2(5x - 8)\) or \((x - 1)^2(5x - 8)\) for the factorisation or \(x = \dfrac{8}{5}\) and \(x = 1\) seen as the roots.

A1: Obtains \(x = 1\) and makes a conclusion as above

(b)(ii) M1: Considers the value of the second derivative either side of \(x = 1\). Do not be too concerned with the interval for the method mark.
(NB \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = (x - 1)(60x - 84)\) so may use this factorised form when considering \(x \lt 1,\ x \gt 1\) for sign change of second derivative)

A1: Fully correct work including a correct \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\) with a reasoned conclusion indicating that the stationary point is a point of inflection. Sufficient reason is e.g. “sign change”/ “> 0, < 0”. If values are given they should be correct (but be generous with accuracy) but also just allow “> 0” and “< 0” provided they are correctly paired. The interval must be where \(x \lt 1.4\)

Alternative 1 for (b)(ii)

M1: Shows that second derivative at \(x = 1\) is zero and then finds the third derivative at \(x = 1\)

A1: Fully correct work including a correct \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\) with a reasoned conclusion indicating that stationary point is a point of inflection. Sufficient reason is “\(\neq 0\)” but must follow a correct third derivative and a correct value if evaluated. For reference \(\left(\dfrac{\mathrm{d}^3y}{\mathrm{d}x^3}\right)_{x=1} = -24\)

Alternative 2 for (b)(ii)

M1: Considers the value of the first derivative either side of \(x = 1\). Do not be too concerned with the interval for the method mark.

A1: Fully correct work with a reasoned conclusion indicating that stationary point is a point of inflection. Sufficient reason is e.g. “same sign”/“both negative”/“< 0, < 0”. If values are given they should be correct (but be generous with accuracy). The interval must be where \(x \lt 1.4\)