if f(x)=sin[pie ^3/3]x + sin[-pie^3/2]x, WHERE [x] IS THE GREATEST INTEGER LESS THAN OR EQUAL TO x , THEN WHAT IS THE VALUE OF f '(pie/2)?

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$$\begin{gathered} \mathrm{We} \mathrm{have} \\ \mathrm{f}\left(\mathrm{x}\right)=\sin \left[\frac{{\mathrm{\pi }}^3}{3}\right]\mathrm{x}+\sin \left[-\frac{{\mathrm{\pi }}^3}{3}\right]\mathrm{x} \\ \mathrm{As} \mathrm{approximate} \mathrm{value} \mathrm{of} \mathrm{\pi } \mathrm{is} 3.1416 \\ {\mathrm{\pi }}^3 \mathrm{will} \mathrm{be} 3.1416\times 3.1416\times 3.1416=31.006 \mathrm{approximately} \\ \mathrm{Hence} \\ \left[\frac{{\mathrm{\pi }}^3}{3}\right]=\left[\frac{31.006}{3}\right]=10 \left[\mathrm{As} \left[\mathrm{x}\right]=10 \mathrm{if} 10\le \mathrm{x}<11\right] \\ \mathrm{and} \\ \left[-\frac{{\mathrm{\pi }}^3}{3}\right]=\left[-\frac{31.006}{3}\right]=-11 \left[\mathrm{As} \left[\mathrm{x}\right]=-11 \mathrm{if} -11\le \mathrm{x}<-10\right] \\ \mathrm{Hence} \mathrm{f}\left(\mathrm{x}\right)=\sin 10\mathrm{x}+\sin \left(-11\right)\mathrm{x} \\ \mathrm{f}\left(\mathrm{x}\right)=\sin 10\mathrm{x}-\sin 11\mathrm{x} \\ \mathrm{Differentiate} \mathrm{with} \mathrm{respect} \mathrm{to} \mathrm{x} \\ \mathrm{f}\text{'}\left(\mathrm{x}\right)=10\cos 10\mathrm{x}-11\cos 11\mathrm{x} \\ \Rightarrow \mathrm{f}\text{'}\left(\frac{\mathrm{\pi }}{2}\right)=10\cos \left(10\frac{\mathrm{\pi }}{2}\right)-11\cos \left(11\frac{\mathrm{\pi }}{2}\right) \\ =10\cos \left(5\mathrm{\pi }\right)-11\times 0 \left[\mathrm{As} \cos \frac{\left(2\mathrm{n}+1\right)\mathrm{\pi }}{2}=0 \mathrm{where} \mathrm{n} \mathrm{is} \mathrm{integer}\right] \\ =10\times \left(-1\right) \left[\mathrm{As} \cos \left(2\mathrm{n}+1\right)\mathrm{\pi }=-1 \mathrm{where} \mathrm{n} \mathrm{is} \mathrm{integer}\right] \\ =-10 \end{gathered}$$

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