If [t] denotes the greatest integer $⩽\mathrm{t}$ then number of points, at which the function $\mathrm{f}\left(\mathrm{x}\right)=4\left|2\mathrm{x}+3\right|+9\left[\mathrm{x}+\frac{1}{2}\right]-12\left[\mathrm{x}+20\right]$  is not differentiable in the open interval  (-20,20) is …………… .

$\begin{array}{l}\mathrm{f}\left(\mathrm{x}\right)=4\left|2\mathrm{x}+3\right|+9\left[\mathrm{x}+\frac{1}{2}\right]-12\left[\mathrm{x}+20\right]\\ \mathrm{x}\in (-20,20)\end{array}$
$\mathrm{f}\left(\mathrm{x}\right)$ is not Diff. at $\mathrm{x}=\mathrm{I}\in \{-19,-18,\dots ..0,\dots 19\}=39$
at $\mathrm{x}=\mathrm{I}+\frac{1}{2},\mathrm{f}\left(\mathrm{x}\right)$ Non diff. at$\left\{-\frac{39}{2},-\frac{37}{2},\dots ,-\frac{3}{2},-\frac{1}{2},\frac{1}{2},\dots ,\frac{39}{2}\right\}$ points
Check at $\mathrm{x}=\frac{-3}{2}$ Discont at $\mathrm{x}=\frac{-3}{2}\therefore \mathrm{N}.\mathrm{R}\left(1\right)$
No. of point of non-differentiability
$=39+40=79$