Consider the function \(f(x) = {x^3} - 3{x^2} + 2x + 2\) . Part of the graph of \(f\) is shown below.

Find \(f'(x)\) .

There are two points at which the gradient of the graph of \(f\) is \(11\). Find the \(x\)-coordinates of these points.

\((f'(x) = )\,\,3{x^2} - 6x + 2\)        (A1)(A1)(A1)     (C3)

Note: Award (A1) for \(3{x^2}\), (A1) for \( - 6x\) and (A1) for \( + 2\).
Award at most (A1)(A1)(A0) if there are extra terms present.

\(11 = 3{x^2} - 6x + 2\)        (M1)

Note: Award (M1) for equating their answer from part (a) to \(11\), this may be implied from \(0 = 3{x^2} - 6x - 9\) .

\((x = )\,\, - 1\,\,,\,\,\,\,(x = )\,\,3\)        (A1)(ft)(A1)(ft)     (C3)

Note: Follow through from part (a).
If final answer is given as coordinates, award at most (M1)(A0)(A1)(ft) for \(( - 1,\,\, - 4)\) and \((3,\,\,8)\) .

Question 15: Differential calculus.

Many candidates correctly differentiated the cubic equation. Most candidates were unable to use differential calculus to find the point where a cubic function had a specified gradient.

Question 15: Differential calculus.

Many candidates correctly differentiated the cubic equation. Most candidates were unable to use differential calculus to find the point where a cubic function had a specified gradient.