(a)     Find in terms of \(a\)

  (i)     the zeros of \(f\) ;

  (ii)     the values of \(x\) for which \(f\) is decreasing;

  (iii)     the values of \(x\) for which \(f\) is increasing;

  (iv)     the range of \(f\) .

(b)     State the concavity of the graph of \(f\) .

(a)

(i)     \(x - a\sqrt x \)    M1

 \(\sqrt x \sqrt x  - a = 0\)     (A1)

  2 \(x = 0\), \(x = {a^2}\)     A1     N2

(ii)     \(f'(x) = 1 - \frac{a}{{2\sqrt x }}\)     A1

  \(f\) is decreasing when \(f' < 0\)     (M1)

  \(1 - \frac{a}{{2\sqrt x }} < 0 \Rightarrow \frac{{2\sqrt x  - a}}{{2\sqrt x }} < 0 \Rightarrow x > \frac{{{a^2}}}{4}\)     A1

(iii)     \(f\) is increasing when \(f' > 0\)

   \(1 - \frac{a}{{2\sqrt x }} > 0 \Rightarrow \frac{{2\sqrt x  - a}}{{2\sqrt x }} > 0 \Rightarrow x > \frac{{{a^2}}}{4}\)     A1

  Note: Award the M1 mark for either (ii) or (iii).

(iv)     minimum occurs at \(x = \frac{{{a^2}}}{4}\)

   minimum value is \(y = - \frac{{{a^2}}}{4}\)     (M1)A1

   hence \(y \geqslant  - \frac{{{a^2}}}{4}\)     A1

[10 marks]

(b)     concave up for all values of \(x\)     R1

[1 mark]

Total [11 marks]

This was generally a well answered question.