Question

Let $f:D\to R$ where $D=[0,1]\cup [2,4]$ be defined by $f(x)=\{\begin{array}{ll}x,& \text{ if }x\in [0,1]\\ 4-x,& \text{ if }x\in [2,4]\end{array}$. 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 $f^{\prime }(\xi )=0$ but Rolle's theorem is not applicabl
• D. $f$ is not continuous in $D$

Answer 1

Answer: (B)

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