Question

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

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

Answer 1

The correct option is C $f$ is differentiable everywhere in $(0,\infty )$

$f:[0,\infty )\to [0,3]$
and $f\left(x\right)=\{\begin{array}{cc}max\{\sin t:0\le t\le x\},& 0\le x\le \pi \\ 2+\cos x,& x>\pi \end{array}$


Clearly $f\left(x\right)$ is continuous everywhere
and $f\left(x\right)$ is differentiable at $x=\frac{\pi }{2}\text{and}x=\pi$
$\therefore f\left(x\right)$ is differentiable everywhere