Question

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

• A. Rolle's theorem is applicable to $f$ in $D$
• B. Rolle's theorem is not applicable to $f$ in $D$
• C. there exists $\xi \in D$ for which $f'(\xi)=0$ but Rolle's theorem is not applicabl
• D. $f$ is not continuous in $D$

Answer 1

Answer: (B)

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