WBJEE · Maths · Application of Derivatives
Let \(R\) be the set of all real numbers and \(f:[-1,1] \rightarrow R\) be defined \(f(x)=\left\{\begin{array}{ll}x \sin \frac{1}{x}, & x \neq 0 \text { . Then } \\ 0, & x=0\end{array}\right.\)
- A \(f\) satisfies the conditions of Rolle's theorem on [-1,1]
- B \(f\) satisfies the conditions of Lagrange's mean value theorem on [-1,1]
- C \(f\) satisfies the conditions of Rolle's theorem on [0,1]
- D \(f\) satisfies the conditions of Lagrange's mean value theorem on [0,1]
Answer & Solution
Correct Answer
(D) \(f\) satisfies the conditions of Lagrange's mean value theorem on [0,1]
Step-by-step Solution
Detailed explanation
Given, \(\quad f(x)=\left\{\begin{array}{cl}x \sin \frac{1}{x}, & x \neq 0 \\ 0, & x=0\end{array}\right.\) Continuity at \(x=0\). \(\mathrm{LHL}=\lim _{x \rightarrow 0^{-}} x \sin \frac{1}{x}=0\) \(\mathrm{RHL}=\lim _{x \rightarrow 0^{*}} x \sin \frac{1}{x}=0\) and…
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