Write down \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).

The curve has a local minimum at the point where \(x = 2\).

Find the value of \(k\).

The curve has a local minimum at the point where \(x = 2\).

Find the value of \(y\) at this local minimum.

\(3{x^2} + k\)     (A1)     (C1)

[1 mark]

\(3{(2)^2} + k = 0\)     (A1)(ft)(M1)

Note: Award (A1)(ft) for substituting 2 in their \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\), (M1) for setting their \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0\).

\(k =  - 12\)     (A1)(ft)     (C3)

Note: Follow through from their derivative in part (a).

[3 marks]

\({2^3} - 12 \times 2\)     (M1)

Note: Award (M1) for substituting 2 and their –12 into equation of the curve.

\( =  - 16\)     (A1)(ft)     (C12)

Note: Follow through from their value of \(k\) found in part (b).

[2 marks]