JEE Main 2022 (Online) 29th July Evening ShiftQuestion 1
If $[t]$ denotes the greatest integer $\le t$, then the number of points, at which the function $f(x)=4|2x+3|+9\left[x+\frac12\right]-12[x+20]$ is not differentiable in the open interval $(-20,20)$, is ________.
Sets, Relations and Function
Answer$79$
Solution
$[x+20]=[x]+20$, so $f(x)=4|2x+3|+9[x+\tfrac12]-12[x]-240$. $[x+\tfrac12]$ jumps (by 1) at half-odd-integers $x=k+\tfrac12$ and $[x]$ jumps at integers; the jumps $9$ and $-12$ are nonzero, so $f$ is discontinuous (not differentiable) at every integer and half-integer. In $(-20,20)$: integers $-19,\ldots,19$ give 39; half-integers $-19.5,\ldots,19.5$ give 40. $|2x+3|$ has a corner at $x=-\tfrac32$, which is already a half-integer counted. Total $=39+40=79$.