If $[t]$ denotes the greatest integer $\leq t$, then the 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 __________.
Answer (integer)
79
Solution
$f(x)=4|2 x+3|+9\left[x+\frac{1}{2}\right]-12[x+20]$
<br/><br/>$=4|2 x+3|+9\left[x+\frac{1}{2}\right]-12[x]-240$
<br/><br/>$f(x)$ is non differentiable at $x=-\frac{3}{2}$
<br/><br/>and $f(x)$ is discontinuous at $\{-19,-18, \ldots ., 18,19\}$ <br/><br/>as well as $\left\{-\frac{39}{2},-\frac{37}{2}, \ldots,-\frac{3}{2},-\frac{1}{2}, \frac{1}{2}, \ldots, \frac{39}{2}\right\}$, <br/><br/>at same point they are also non differentiable
<br/><br/>$$
\begin{aligned}
\therefore & \text { Total number of points of non differentiability } \\
&=39+40 \\
&=79
\end{aligned}
$$
About this question
Subject: Mathematics · Chapter: Limits, Continuity and Differentiability · Topic: Limits and Standard Results
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