Statement - 1: Let $f(x)=[3+4 \sin x]$, where [.] denotes the greatest integer function. The number of discontinuities of $f(x)$ in $[\pi, 2 \pi]$ is 6.
Statement - 2: The range of $f$ is $\{-1,0,1,2,3\}$.
Answer & explanation
Correct answer: option 4
We have,
$-1 \leq \sin x \leq 0$ for all $x \in[\pi, 2 \pi]$
$\Rightarrow -4 \leq 4 \sin x \leq 0$ for all $x \in[\pi, 2 \pi]$
$\Rightarrow -1 \leq 4 \sin x+3 \leq 3$ for all $x \in[\pi, 2 \pi]$
$\Rightarrow f(x)=[4 \sin x+3]$ assumes values -1, 0, 1, 2 and 3 when $x \in[\pi, 2 \pi] $
⇒ Range f = {-1, 0, 1, 2, 3}
So, statement- 2 is true.
Clearly, there are eight points of discontinuity of $f(x)$ in $[\pi, 2 \pi]$. So statement- 1 is not true.