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Maths · Applications of derivatives: Rate of change of quantities, monotonic-Increasing and decreasing functions, Maxima and minima of functions of one variable

Let be the set of all real numbers and \rightarrow \boldsymbol{R} \) be defined

Let \( R \) be the set of all real numbers and \( \boldsymbol{f}:[-1,1] \rightarrow \boldsymbol{R} \) be defined by \( \boldsymbol{f}(\boldsymbol{x})= \) \( \left\{\begin{array}{l}x \sin \frac{1}{x} ; \quad x \neq 0 \\ 0 ; \quad x=0\end{array}\right. \)

  • 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]

Step-by-step solution

f is continuous on [0,1] and differentiable on (0,1) because f'(x) exists for all x≠0. Hence, Lagrange's mean value theorem conditions are satisfied on [0,1]. Options A and B fail because f is not differentiable at 0, which is in the interior. Option C fails because f(0)≠f(1).
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