Problem source

\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}
\begin{questionparts}
\item By considering $\f(0)$ and $\f'(x)$, show that $\f(x)>0$
for $0<x<\tfrac{1}{2}\pi$.
\item Show similarly that $\g(x)>0$ for $0<x<\tfrac{1}{2}\pi$.
\item Show that $\h(x)>0$ for $0<x<\tfrac{1}{4}\pi$, and hence that
\[x(\sin^2x+3\cos^2x)-3\sin x\cos x>0\] for $0<x<\tfrac{1}{4}\pi$.
\item By considering $\displaystyle {{\rm F}'(x)\over {\rm F}(x)}$,
 show that ${\rm F}'(x)<0$ for $0<x<\tfrac{1}{4}\pi$.
\end{questionparts}