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  • Consider the function                <img alt="f(x)=\left\{\begin{array}{cc} x \sin \frac{\pi}{x}, & \text { for } x>0 \\ 0, & \tex

Consider the function 

              f(x)=\left\{\begin{array}{cc} x \sin \frac{\pi}{x}, & \text { for } x>0 \\ 0, & \text { for } x=0 \end{array}\right.

then the number of points in (0,1), where the derivative f^{\prime}(x) vanishes, is

 

Option: 1

0


Option: 2

1


Option: 3

2


Option: 4

infinite


Answers (1)

best_answer

f(x) \text { vanishes at points, where } \sin \frac{\pi}{x}=0

i.e., \frac{\pi}{x}=k \pi, k=1,2,3,4, \ldots

Hence, x=\frac{1}{k}. Also , f^{\prime}(x)=\sin \frac{\pi}{x}-\frac{\pi}{x} \cos \frac{\pi}{x}, \text { if } x \neq 0

Since, the function has a derivative at any interior point of the interval $(0,1)$, also continuous in [0,1] and f(0)=f(1), hence Rolle's theorem is applicable to any one of the interval \left[\frac{1}{2}, 1\right],\left[\frac{1}{3}, \frac{1}{2}\right], \ldots,\left[\frac{1}{k+1}, \frac{1}{k}\right], hence \exists some c in each of these interval, where f^{\prime}(c)=0\: \Rightarrow infinite points.

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Rishi

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