3
$\begingroup$

Prove that $f(x) = 2^{-x}+2^{-1/x}$ has the unique local maximum $(1,1)$ for $x>0$. Do not use computer software.

Proving that $(1,1)$ is a maximum is easy, but I'm having trouble with the uniqueness part. The first derivative has an unclear number of zeroes because of its mixed logarithmic/polynomial form.

I've reduced it to showing that $x = 2^{(x-1/x)/2}$ has a unique fixed point on $(0,1)$, but this has proved difficult as well. None of the typical unique fixed point theorems seem to apply.

I'm sure that certain iterative methods could show uniqueness, but I'm not sure how to use these without resorting to software and approximations.

$\endgroup$
3

1 Answer 1

1
$\begingroup$

Not as pleasing as the AM-GM inequality, but since $f$ and its derivatives are all continuous except the discontinuity at $x=0$:

  1. you can differentiate $f$, then analytically show $f'$ is zero at x=1, as well as somewhere around 0.2 (here using the intermediate value theorem].

  2. obtain $f''$ and show it is positive at $x\approx 0.2$ and negative at $x=1.$

  3. conclude $x=1$ is a unique local maximum of this function

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .