The number of critical points of the function$f(x) = \left\{ \begin{matrix} \left| \frac{sinx}{x} \r

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The number of critical points of the function$f(x) = \left\{ \begin{matrix} \left| \frac{sinx}{x} ight|, & x 
eq 0 \ 1, & x = 0 \end{matrix} ight.\ $in the interval $( - 2\pi,2\pi)$ is equal to:

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Sol. [Key Concept: Solve $f'(x) = 0 \Rightarrow \ tan\ x = x$ (3 roots) and add the points where $f'$ fails to exist.]$\ \lim_{x \rightarrow 0}\mspace{2mu}\left| \frac{sinx}{x} \right| = 1 = f(0) \rightarrow f(x)$ is continuousNow $\frac{d}{dx}\left( \frac{sinx}{x} \right) = \frac{xcosx - sinx}{x^{2}}$$$\Rightarrow f^{'}(x) = 0 \Rightarrow tanx = x $$3 solution in ( $- 2\pi,2\pi$ )also $f^{'}(0)$ does not exist at $x = - \pi,\pi$total 5 points

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