Which of the following is-are true- For fx - ln ln x-ln x

Let f(x) = ln (ln x)/ln x f'(x)= 1 ln (ln x)/x(ln x)^2 0 ∀ x e^e So f(x) is increasing in (1, e^e) and decreasing in (e^e, ∞) So x y ⇒ (lnx)^lny (lny)^lnx∀ x ...

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Standard XIII

Mathematics

Monotonicity in an Interval

Which of the ...

Question

Which of the following is/are true, For $f (x) = \frac{l n (l n x)}{l n x}$

(l n 2.1)^{l n 2.2} > (l n 2.2)^{l n 2.1}

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(l n 4)^{l n 5} < (l n 5)^{l n 4}

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(l n 30)^{l n 31} < (l n 31)^{l n 30}

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(l n 28)^{l n 30} < (l n 30)^{l n 28}

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Solution

The correct option is B $(l n 4)^{l n 5} < (l n 5)^{l n 4}$
Let f(x) = $\frac{l n (l n x)}{l n x}$
f'(x)= $\frac{1 - l n (l n x)}{x (l n x)^{2}} < 0 \forall x > e^{e}$
So f(x) is increasing in (1, $e^{e}$ ) and decreasing in $(e^{e}, \infty)$
So x $> y \Rightarrow (l n x)^{l n y} < (l n y)^{l n x} \forall x, y ϵ (e^{e}, \infty)$
and x $< y \Rightarrow (l n x)^{l n y} < (l n y)^{l n x} \forall x, y ϵ (1, e^{e})$

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

The correct option is B (ln 4)ln5 < (ln 5)ln4Let f(x) = ln(lnx)lnx f'(x)= 1−ln(lnx)x(lnx)2<0∀x>ee So f(x) is increasing in (1, ee) and decreasing in (ee,∞) So x >y⇒(lnx)lny<(lny)lnx∀x,yϵ(ee,∞) and x <y⇒(lnx)lny<(lny)lnx∀x,yϵ(1,ee)

Which of the following is/are true, For $f (x) = \frac{l n (l n x)}{l n x}$