How to demonstrate that 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.

1 Answer
Apr 3, 2017

See below.

Explanation:

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.