Limits, Continuity & Differentiability
Non-Differentiability of Floor of Sinusoidal Function
nta_pyq_2023_apr
Grade 12
Question:
Let $a\in\mathbb{Z}$ and $[t]$ be the greatest integer $\leq t$, then the number of points, where the function $f(x)=[a+13\sin x]$, $x\in(0,\pi)$ is not differentiable, is ____________.
Step-by-Step Solution
Key Concept: Since $a$ is an integer, $[a+13\sin x]=[13\sin x]+a$. The function $g(x)=13\sin x$ ranges from $0$ to $13$ on $(0,\pi)$, being discontinuous (hence non-differentiable) at each integer value of $13\sin x$.
$g(x)=13\sin x$ crosses integer values $1,2,\ldots,12$ twice each and $13$ once. Total $=25$ points.
Correct Answer: 25