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  4. If [t] denotes the greatest integer ≤ t, then number of points, at which the function f(x)=4|2 x+3|+9[x+(1/2)]-12[x+20] is not differentiable in the open interval (-20,20), is

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

JEE MainJEE Main 2022Continuity and Differentiability

Solution:

$ f(x)=4|2 x+3|+9\left[x+\frac{1}{2}\right]-12[x+20]$
$ x \in(-20,20)$
$f ( x )$ is not Diff. at
$x = I \in\{-19,-18, \ldots .0, \ldots 19\}=39$
at $x=I+\frac{1}{2}, f(x)$ Non diff. at 39 points
Check at $x=\frac{-3}{2}$ Discount at $x=\frac{-3}{2}$
$\therefore$ N. R(1)
No. of point of non-differentiabilty
$=39+39+1=79$