JEE Main 2022 — Continuity and Differentiability Question with Solution

Given,

$f\left(x\right)=4\left|2x+3\right|+9\left[x+\frac{1}{2}\right]-12\left[x+20\right]$

$x\in \left(-20,20\right)$

Now we know that $\left[x\right]$ is non-differentiable when $x\in I$

So, $12\left[x+20\right]$ is not differentiable at $x=I\in \left\{-19,-18,\dots 0,\dots 19\right\}=39$

Now  $9\left[x+\frac{1}{2}\right]$ is not differentiable when $x=I+\frac{1}{2}$

So, here also total non-differentiable points are $39$

Now checking modulus function

We know that modulus function are non-differentiable at point where modulus function is zero,

So at $x=\frac{-3}{2}$ modulus function is zero

Also $f\left(x\right)$  at $\mathrm{x}=\frac{-3}{2}$ is not continuous and non-differentiable,

So, number of point of non-differentiability$=39+39+1=79$