Limit Continuity and Differentiability 7 Question 25
25. For a real number $y$, let $[y]$ denotes the greatest integer less than or equal to $y$. Then, the function $f(x)=\frac{\tan \pi[(x-\pi)]}{1+[x]^{2}}$ is
$(1981,2 M)$
(a) discontinuous at some $x$
(b) continuous at all $x$, but the derivative $f^{\prime}(x)$ does not exist for some $x$
(c) $f^{\prime}(x)$ exists for all $x$, but the derivative $f^{\prime \prime}(x)$ does not exist for some $x$
(d) $f^{\prime}(x)$ exists for all $x$
Objective Questions II
(One or more than one correct option)
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Answer:
Correct Answer: 25. $(a, b)$
Solution:
- Given, $f(x)=\log _x(\log x)$
$$ \therefore \quad f(x)=\frac{\log (\log x)}{\log x} $$
On differentiating both sides, we get
$$ \begin{aligned} f^{\prime}(x) & =\frac{(\log x) \frac{1}{\log x} \cdot \frac{1}{x}-\log (\log x) \cdot \frac{1}{x}}{(\log x)^{2}} \\ \therefore \quad f^{\prime}(e) & =\frac{1 \cdot \frac{1}{1} \cdot \frac{1}{e}-\log (1) \cdot \frac{1}{e}}{(1)^{2}} \\ \Rightarrow \quad f^{\prime}(e) & =\frac{1}{e} \end{aligned} $$
