Let $$f_m(x) = \frac{x}{2} \left[ \sin x - \frac{\sin 2x}{2} + \frac{\sin 3x}{3} - \ldots + (-1)^{m+1} \frac{\sin mx}{m} \right].$$ By considering \(df_m(x)/dx\), or otherwise, show that $$(-1)^m f_m(x) > 0$$ for \(0 < x < \pi/(2m+1)\). Show also that $$(-1)^m f_m\left(\frac{\pi}{m+\frac{1}{3}}\right) < 0.$$