(i)     Suppose that $f(0)=0$ and that, for $x\ne 0$,

${\displaystyle 0\le \frac{f(x)}{x}\le 1}$

. Show that

${\displaystyle -\frac{1}{2}\le {\int }_{-1}^{1}f(x)\hspace{0.5em}\mathrm{dx}\le \frac{1}{2}.}$

(ii)     Suppose that $f(0)=0$ and that, for $x\ne 0$,

${\displaystyle 0\le \frac{f(x)}{x^2}\le 1}$

. Show that

${\displaystyle 0\le {\int }_{-1}^{1}f(x)\hspace{0.5em}\mathrm{dx}\le \frac{2}{3}.}$

(iii)     Generalize (i) and (ii) to the case where $f(0)=0$ and, for $x\ne 0$,

${\displaystyle 0\le \frac{f(x)}{x^n}\le 1}$

where $n$ is a positive integer.