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Let \( f(x)=x+\log _{e} x-x \log _{0} x, x \in(0, \infty) \).
Column 1 contains information about zeros of \( f(x), f(x) \) and \( f^{\prime}(x) \).
Column 2 contains information about the limiting behavior of \( \mathrm{f}(\mathrm{x}), \mathrm{f}(\mathrm{x}) \) and \( \mathrm{f}^{\prime}(\mathrm{x}) \) at infinity.
\( \mathrm{P} \)
Column 3 contains information about increasing/decreasing nature of \( f(x) \) and \( f(x) \).
W
Column 1
Column 2 (i) \( \lim _{x \rightarrow \infty} f(x)=0 \) (ii) \( \lim _{x \rightarrow \infty} f(x)=-\infty \) (iii) \( \lim _{x \rightarrow \infty} f(x)=-\infty \) (iv) \( \lim _{x \rightarrow \infty} f(x)=0 \)
Column 3
\( \begin{array}{lll}\text { (I) } f(x)=0 \text { for some } x \in\left(1, e^{2}\right) & \text { (i) } \lim _{x \rightarrow x} f(x)=0 & \text { (P) } f \text { is increasing in }(0,1)\end{array} \)
\( \begin{array}{lll}\text { (II) } \mathrm{f}^{\prime}(\mathrm{x})=0 \text { for some } \mathrm{x} \in(\mathrm{1}, \mathrm{e}) & \text { (ii) } \lim _{x \rightarrow \infty} \mathrm{f}(\mathrm{x})=-\infty & \left.\text { (Q) } \mathrm{f} \text { is decreasing in (e, } \mathrm{e}^{2}\right)\end{array} \)
(III) \( \mathrm{f}^{\prime}(\mathrm{x})=0 \) for some \( \mathrm{x} \in(0,1) \)
(R) \( f \) is increasing in \( (0,1) \)
(IV) \( \mathrm{f}^{\prime}(\mathrm{x})=0 \) for some \( \mathrm{x} \in(1, \mathrm{e}) \)
(S) \( f \) is decreasing in \( \left(e, e^{2}\right) \)
Which of the following options is the only CORRECT
combination?
[JEE Advanced -2017]
(A) (I) (i) (P)
(B) (II) (ii) (Q)
(C) (III) (iii) (R)
(D) (IV) (iv) (S)
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