How to demonstrate that ${\sqrt{2}}^{\sqrt{3} - 1} > {\sqrt{3}}^{\sqrt{2} - 1}$ $sqrt(2)^(sqrt(3)-1)>sqrt(3)^(sqrt(2)-1)$ ?We know that $g:(1,+oo)->RR,g(x)=$ $(lnx)/(x-1)$ ,and $g$ is decreasing.

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Answer 1 · 2017-04-03T17:33:17

See below.

We have to compare

$(sqrt(a))^(sqrt(b)-1)$ and $(sqrt(b))^(sqrt(a)-1)$

Applying $log$ to both sides

$(sqrt b - 1)log(sqrt(a)) > (sqrta -1)log(sqrt(b))$ or

$log(sqrt(a))/(sqrta - 1) > log(sqrtb)/(sqrtb - 1)$

The function $g(x) = log(x)/(x-1)$ is strictly monotonic decreasing so

if $a > b$ then $g(b) > g(a)$ . Here $a = sqrt(2) < b = sqrt(3)$ then

$g(2) > g(3)$ as proposed.

How to demonstrate that sqrt(2)^(sqrt(3)-1)>sqrt(3)^(sqrt(2)-1)√2√3−1>√3√2−1sqrt(2)^(sqrt(3)-1)>sqrt(3)^(sqrt(2)-1)?We know that g:(1,+oo)->RR,g(x)=g:(1,+oo)->RR,g(x)=(lnx)/(x-1)(lnx)/(x-1),and gg is decreasing.