Expand the expression \(x(2{x^3} - 1)\).

Differentiate \(f(x) = x(2{x^3} - 1)\).

Find the \(x\)-coordinate of the local minimum of the curve \(y = f(x)\).

\(2{x^4} - x\)     (A1)(A1)     (C2)

Note: Award (A1) for \(2{x^4}\), (A1) for \( - x\).

[2 marks]

\(8{x^3} - 1\)     (A1)(ft)(A1)(ft)     (C2)

Note: Award (A1)(ft) for \(8{x^3}\), (A1)(ft) for \(–1\). Follow through from their part (a).

     Award at most (A1)(A0) if extra terms are seen.

[2 marks]

\(8{x^3} - 1 = 0\)     (M1)

Note: Award (M1) for equating their part (b) to zero.

\((x = )\frac{1}{2}{\text{ (0.5)}}\)     (A1)(ft)     (C2)

Notes: Follow through from part (b).

     \(0.499\) is the answer from the use of trace on the GDC; award (A0)(A0).

     For an answer of \((0.5, –0.375)\), award (M1)(A0).

[2 marks]

A surprising number of candidates were unable to correctly expand the expression given in part (a). Most candidates were able to differentiate their function but a considerable number were unable to find the x-coordinate of the minimum point. Candidates must read the questions correctly as answers giving ordered pairs were not awarded the final mark. A number of candidates did not use calculus to determine the local minimum but graphed the function, often achieving full marks for part (c), even when parts (b) or (a) were incorrect or left blank.

A surprising number of candidates were unable to correctly expand the expression given in part (a). Most candidates were able to differentiate their function but a considerable number were unable to find the x-coordinate of the minimum point. Candidates must read the questions correctly as answers giving ordered pairs were not awarded the final mark. A number of candidates did not use calculus to determine the local minimum but graphed the function, often achieving full marks for part (c), even when parts (b) or (a) were incorrect or left blank.

A surprising number of candidates were unable to correctly expand the expression given in part (a). Most candidates were able to differentiate their function but a considerable number were unable to find the x-coordinate of the minimum point. Candidates must read the questions correctly as answers giving ordered pairs were not awarded the final mark. A number of candidates did not use calculus to determine the local minimum but graphed the function, often achieving full marks for part (c), even when parts (b) or (a) were incorrect or left blank.