If $\mathrm{f}\left(\mathrm{x}\right)=2(7\cos \mathrm{x}+24\sin \mathrm{x})(7\sin \mathrm{x}-24\cos \mathrm{x})$  for every  $\mathrm{x}\in \mathrm{R}$ then maximum value of f (x) is

$\mathrm{f}\left(\mathrm{x}\right)=2(7\cos \mathrm{x}+24\sin \mathrm{x})(7\sin \mathrm{x}-24\cos \mathrm{x})$

$\text{ Let }\mathrm{rcos}\mathrm{\theta }=7;\mathrm{rsin}\mathrm{\theta }=24$

$\begin{array}{l}\therefore {\mathrm{r}}^2=625;\tan \mathrm{\theta }=\frac{24}{7}\\ \therefore \mathrm{f}\left(\mathrm{x}\right)=2\mathrm{rcos}(\mathrm{x}-\mathrm{\theta })\times \mathrm{rsin}(\mathrm{x}-\mathrm{\theta })\\ ={\mathrm{r}}^2(\sin 2(\mathrm{x}-\mathrm{\theta }\left)\right)\\ \mathrm{f}(\mathrm{x}{)}_{max}={25}^2\end{array}$