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Question

Which of the following is/are true, For f(x) = ln (ln x)ln x

A
(ln 2.1)ln2.2 > (ln 2.2)ln2.1
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B
(ln 4)ln5 < (ln5)ln4
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C
(ln30)ln31 < (ln31)ln30
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D
(ln28)ln30 < (ln30)ln28
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Solution

The correct option is B (ln 4)ln5 < (ln5)ln4
Let f(x) = ln(lnx)lnx
f'(x)= 1ln(lnx)x(lnx)2<0x>ee
So f(x) is increasing in (1, ee) and decreasing in (ee,)
So x >y(lnx)lny<(lny)lnxx,yϵ(ee,)
and x <y(lnx)lny<(lny)lnxx,yϵ(1,ee)

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