Problems

Throughout this question, consider only \(x > 0\).

1. Let \[\mathrm{g}(x) = \ln\left(1 + \frac{1}{x}\right) - \frac{x+c}{x(x+1)}\] where \(c \geqslant 0\).
   1. Show that \(y = \mathrm{g}(x)\) has positive gradient for all \(x > 0\) when \(c \geqslant \frac{1}{2}\).
   2. Find the values of \(x\) for which \(y = \mathrm{g}(x)\) has negative gradient when \(0 \leqslant c < \frac{1}{2}\).
2. It is given that, for all \(c > 0\), \(\mathrm{g}(x) \to -\infty\) as \(x \to 0\). Sketch, for \(x > 0\), the graphs of \[y = \mathrm{g}(x)\] in the cases
   1. \(c = \frac{3}{4}\),
   2. \(c = \frac{1}{4}\).
3. The function \(\mathrm{f}\) is defined as \[\mathrm{f}(x) = \left(1 + \frac{1}{x}\right)^{x+c}.\] Show that, for \(x > 0\),
   1. \(\mathrm{f}\) is a decreasing function when \(c \geqslant \frac{1}{2}\);
   2. \(\mathrm{f}\) has a turning point when \(0 < c < \frac{1}{2}\);
   3. \(\mathrm{f}\) is an increasing function when \(c = 0\).

1. By considering the Maclaurin series for \(\mathrm{e}^x\), show that for all real \(x\), \[\cosh^2 x \geqslant 1 + x^2.\] Hence show that the function \(\mathrm{f}\), defined for all real \(x\) by \(\mathrm{f}(x) = \tan^{-1} x - \tanh x\), is an increasing function. Sketch the graph \(y = \mathrm{f}(x)\).
2. Function \(\mathrm{g}\) is defined for all real \(x\) by \(\mathrm{g}(x) = \tan^{-1} x - \frac{1}{2}\pi \tanh x\).
   1. Show that \(\mathrm{g}\) has at least two stationary points.
   2. Show, by considering its derivative, that \((1+x^2)\sinh x - x\cosh x\) is non-negative for \(x \geqslant 0\).
   3. Show that \(\dfrac{\cosh^2 x}{1+x^2}\) is an increasing function for \(x \geqslant 0\).
   4. Hence or otherwise show that \(\mathrm{g}\) has exactly two stationary points.
   5. Sketch the graph \(y = \mathrm{g}(x)\).

In this question, you may assume without proof that any function \(\f\) for which \(\f'(x)\ge 0\) is increasing; that is, \(\f(x_2)\ge \f(x_1)\) if \(x_2\ge x_1\,\).

1. 1. Let \(\f(x) =\sin x -x\cos x\). Show that \(\f(x)\) is increasing for \(0\le x \le \frac12\pi\,\) and deduce that \(\f(x)\ge 0\,\) for \(0\le x \le \frac12\pi\,\).
   2. Given that \(\dfrac{\d}{\d x} (\arcsin x) \ge1\) for \(0\le x< 1\), show that \[ \arcsin x\ge x \quad (0\le x < 1). \]
   3. Let \(\g(x)= x\cosec x\, \text{ for }0< x < \frac12\pi\). Show that \(\g\) is increasing and deduce that \[ ({\arcsin x})\, x^{-1} \ge x\,{\cosec x} \quad (0 < x < 1). \]
2. Given that $\dfrac{\d}{\d x} (\arctan x)\le 1\text{ for }x\ge 0$, show by considering the function \(x^{-1} \tan x\) that \[ (\tan x)( \arctan x) \ge x^2 \quad (0< x < \tfrac12\pi). \]

\begin{eqnarray*} {\rm f}(x)&=& \tan x-x,\\ {\rm g}(x)&=& 2-2\cos x-x\sin x,\\ {\rm h}(x)&=& 2x+x\cos 2x-\tfrac{3}{2}\sin 2x,\\ {\rm F}(x)&=& {x(\cos x)^{1/3}\over\sin x}. \end{eqnarray*} \vspace{1mm}

1. By considering \(\f(0)\) and \(\f'(x)\), show that \(\f(x)>0\) for \(0
2. Show similarly that \(\g(x)>0\) for \(0
3. Show that \(\h(x)>0\) for \(00\] for \(0
4. By considering \(\displaystyle {{\rm F}'(x)\over {\rm F}(x)}\), show that \({\rm F}'(x)<0\) for \(0