QJEEM21Mae41e: JEE Main 2021 Mathematics - Class 12 JEEM21Mae41e Solution: Let f be a real valued function, defined on R { 1, 1} and given …

Question

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Let f be a real valued function, defined on R

-

{

-

1, 1} and given by

f(x) = 3 log_e

\left| {{{x - 1} \over {x + 1}}} \right| - {2 \over {x - 1}}

.

Then in which of the following intervals, function f(x) is increasing?

A ( $-$ $\infty$ , $-$ 1) $\cup$ $\left( {[{1 \over 2},\infty ) - \{ 1\} } \right)$
B ( $-$ $\infty$ , $\infty$ ) $-$ { $-$ 1, 1)
C ( $-$ $\infty$ , ${{1 \over 2}}$ ] $-$ { $-$ 1}
D ( $-$ 1, ${{1 \over 2}}$ ]

Accepted Answer

f(x) = 3 log_e

\left| {{{x - 1} \over {x + 1}}} \right| - {2 \over {x - 1}}

f'(x) = {{3(x + 1)} \over {x - 1}} \times {{(x + 1) - (x - 1)} \over {{{(x + 1)}^2}}} + {2 \over {{{(x - 1)}^2}}} &gt; 0

= {6 \over {{x^2} - 1}} + {2 \over {{{(x - 1)}^2}}} &gt; 0

= {{2(3(x - 1) + (x + 1))} \over {{{(x - 1)}^2}(x + 1)}} = {{4(2x - 1)} \over {{{(x - 1)}^2}(x + 1)}} &gt; 0

\Rightarrow x \in ( - \infty , - 1) \cup \left[ {{1 \over 2},\infty ) - \{ 1\} } \right.

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