WBJEE · Maths · Application of Derivatives
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 \(\mathrm{D}\)
- 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 \(D\)
Answer & Solution
Correct Answer
(B) Rolle's theorem is not applicable to \(\mathrm{f}\) in \(\mathrm{D}\)
Step-by-step Solution
Detailed explanation
\(f(x)=\left\{\begin{array}{ll}x, & x \in[0,1] \\ 4-x, & x \in[2,4]\end{array}\right.\) \(f(x)\) is increasing in \([0,1]\) and \(f(x)\) is decreasing in \([2,4]\) \(\therefore\) Rolle's theorem is not applicable to \(\mathrm{f}\) in \(\mathrm{D}\).
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