Let f: D rightarrow R where D=[0,1] cup [2,4] | vedclass

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(B) Rolle's theorem requires the function $f(x)$ to be defined on a closed interval $[a, b]$, continuous on $[a, b]$, and differentiable on $(a, b)$.H

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Rolle's theorem is not applicable to $f$ in $D$

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(B) Rolle's theorem requires the function $f(x)$ to be defined on a closed interval $[a, b]$, continuous on $[a, b]$, and differentiable on $(a, b)$.Here, the domain $D = [0, 1] \cup [2, 4]$ is not a single closed interval. It is a union of two disjoint closed intervals.For Rolle's theorem to be applicable, the domain must be a single connected closed interval $[a, b]$.Since the domain $D$ is disconnected, Rolle's theorem is not applicable to $f$ in $D$.

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,

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