Question

Let $f:[0, \infty) \rightarrow[0,3]$ be a function defined by
$f(x)=\begin{cases}\max \{\sin t: 0 \leq t \leq x\}, & 0 \leq x \leq \pi \\2+\cos x, & x>\pi \end{cases}$
Then which of the following is true?

• A. $f$ is continuous everywhere but not differentiable exactly at one point in $(0, \infty)$
• B. $f$ is differentiable everywhere in $(0, \infty)$
• C. $f$ is not continuous exactly at two points in $(0, \infty)$
• D. $f$ is continuous everywhere but not differentiableexactly at two points in $(0, \infty)$

Answer 1

Answer: (B)

Graph of $\max \{\sin t: 0 \leq t \leq x\}$ in $x \in[0, \pi]$

So graph of
$f(x)=\begin{cases}\max \{\sin t: 0 \leq t \leq x\}, & 0 \leq x \leq \pi \\2+\cos x, & x> h\end{cases}$

$f$ is differentiable everywhere in $(0, \infty)$