Let $f: D \rightarrow R$ where $D=[0,1] \cup[2,4]$ be defined by $f(x)=\left\{\begin{array}{ll}x, & \text { if } x \in[0,1] \\ 4-x, & \text { if } x \in[2,4]\end{array}\right.$. Then,

• A

  Rolle's theorem is applicable to $\mathrm{f}$ in $\mathrm{D}$
• Correct answer
  B

  Rolle's theorem is not applicable to $\mathrm{f}$ in $\mathrm{D}$
• C

  there exists $\xi \in \mathrm{D}$ for which $\mathrm{f}^{\prime}(\xi)=0$ but Rolle's theorem is not applicable
• D

  $f$ is not continuous in $D$

Solution

$f(x)=\left\{\begin{array}{ll}x, & x \in[0,1] \\ 4-x, & x \in[2,4]\end{array}\right.$ $f(x)$ is increasing in $[0,1]$ and $f(x)$ is decreasing in $[2,4]$ $\therefore$ Rolle's theorem is not applicable to $\mathrm{f}$ in $\mathrm{D}$.

Hence, the correct answer is option: 'B'.