Let \(f(x) = \frac{{1 - x}}{{1 + x}}\) and \(g(x) = \sqrt {x + 1} \), \(x > - 1\).

Find the set of values of \(x\) for which \(f'(x) \leqslant f(x) \leqslant g(x)\) .

\(f'(x) = \frac{{ - 2}}{{{{\left( {1 + x} \right)}^2}}}\)     M1A1

Note: Alternatively, award M1A1 for correct sketch of the derivative.

find at least one point of intersection of graphs     (M1)

\(y = f(x)\) and \(y = f'(x)\) for \(x = \sqrt 3 \) or \(1.73\)     (A1)

\(y = f(x)\) and \(y = g(x)\) for \(x = 0\)     (A1)

forming inequality \(0 \leqslant x \leqslant \sqrt 3 \) (or \(0 \leqslant x \leqslant 1.73\))     A1A1     N4

Note: Award A1 for correct limits and A1 for correct inequalities.

[7 marks]

Most students were able to find the derived function correctly, although attempts to solve the inequality algebraically were often unsuccessful. This was a question where students prepared in good use of GDC were able to easily obtain good marks.