Write down the equation of the vertical asymptote.

Find \(f'(x)\) .

Find the gradient of the graph of the function at \(x = - 1\).

Using your answer to part (c), decide whether the function \(f(x)\) is increasing or decreasing at \(x = - 1\). Justify your answer.

Sketch the graph of \(f(x)\) for \( - 10 \leqslant x \leqslant 10\) and \( - 20 \leqslant y \leqslant 20\) .

\({{\text{P}}_1}\) is the local maximum point and \({{\text{P}}_2}\) is the local minimum point on the graph of \(f(x)\) .

Using your graphic display calculator, write down the coordinates of

(i)     \({{\text{P}}_1}\) ;

(ii)    \({{\text{P}}_2}\) .

Using your sketch from (e), determine the range of the function \(f(x)\) for \( - 10 \leqslant x \leqslant 10\) .

\(x = 0\)     (A1)(A1)

Note: Award (A1) for \(x = {\text{constant}}\), (A1) for \(0\).

[2 marks]

\(f'(x) = 1.5 - \frac{6}{{{x^2}}}\)     (A1)(A1)(A1)

Notes: Award (A1) for \(1.5\), (A1) for \( - 6\), (A1) for \({x^{ - 2}}\) . Award (A1)(A1)(A0) at most if any other term present.

[3 marks]

\(1.5 - \frac{6}{{( - 1)}}\)     (M1)

\( =  - 4.5\)     (A1)(ft)(G2)

Note: Follow through from their derivative function.

[2 marks]

Decreasing, the derivative (gradient or slope) is negative (at \(x = - 1\))     (A1)(R1)(ft)

Notes: Do not award (A1)(R0). Follow through from their answer to part (c).

[2 marks]

     (A4)
Notes: Award (A1) for labels and some indication of scales and an appropriate window.

Award (A1) for correct shape of the two unconnected, and smooth branches.

Award (A1) for the maximum and minimum points in the approximately correct positions.

Award (A1) for correct asymptotic behaviour at \(x = 0\) .

Notes: Please be rigorous.

The axes need not be drawn with a ruler.

The branches must be smooth and single continuous lines that do not deviate from their proper direction.

The max and min points must be symmetrical about point \((0{\text{, }}4)\) .

The \(y\)-axis must be an asymptote for both branches.

[4 marks]

(i)     \(( - 2{\text{, }} - 2)\) or \(x = - 2\), \(y = - 2\)     (G1)(G1)

(ii)    \((2{\text{, }}10)\) or \(x = 2\), \(y = 10\)     (G1)(G1)

[4 marks]

\(\{  - 2 \geqslant y\} \) or \(\{ y \geqslant 10\} \)     (A1)(A1)(ft)(A1)

Notes: (A1)(ft) for \(y > 10\) or \(y \geqslant 10\) . (A1)(ft) for \(y < - 2\) or \(y \leqslant - 2\) . (A1) for weak (non-strict) inequalities used in both of the above. Follow through from their (e) and (f).

[3 marks]

Part a) was either answered well or poorly.

Most candidates found the first term of the derivative in part b) correctly, but the rest of the terms were incorrect.

The gradient in c) was for the most part correctly calculated, although some candidates substituted incorrectly in \(f(x)\) instead of in \(f'(x)\) .

Part d) had mixed responses.

Lack of labels of the axes, appropriate scale, window, incorrect maximum and minimum and incorrect asymptotic behaviour were the main problems with the sketches in e).

Part f) was also either answered correctly or entirely incorrectly. Some candidates used the trace function on the GDC instead of the min and max functions, and thus acquired coordinates with unacceptable accuracy. Some were unclear that a point of local maximum may be positioned on the coordinate system “below” the point of local minimum, and exchanged the pairs of coordinates of those points in f(i) and f(ii).

Very few candidates were able to identify the range of the function in (g) irrespective of whether or not they had the sketches drawn correctly.