In Problem 1, we asked:
Without using a calculator, work out which is larger, or .
First let us think about strictly monotone increasing functions. A function is strictly monotone increasing if if and only if . Now, is strictly monotone increasing, therefore (why?) so is its inverse, for . Now let’s go back to the problem.
Suppose we want to prove that , for . Since is strictly monotone increasing, is logically equivalent to , i.e. , which is the same as , which is the same as . But since is strictly monotone increasing, this is the same as . So what have we shown? That if and only if . This is good, as now we can use calculus.
Consider . Then by the chain rule (check!),
So the only critical point of is at . This clearly is a maximum, as and as (by l’Hôpital; check!) But then for any and so in particular , which as we have shown is equivalent to . They are pretty close! and .
(Brownie points to Michael Nolan, but no cigar.)