You are here:

Advanced Math/Extrema of f(x) = ln(x) * ln(1-x)


I wonder how one can give a rigorous argument about the uniqueness of extremum of the function above.

This question arises when minimizing the probability of false positives,
p(k) = (1 - [1 - 1/m]^(k*n))^k /approx (1 - e^(-(k*n)/m))^k ,
of Bloom filters; see for background.

All of the interenet resources dealing with the subject I could find, e.g.,
seem to brush the question of uniqueness aside.


The function f(x) = ln(x) ln(1-x) has derivative:

f'(x) = log(1-x)/x - log(x)/(1-x)

It is clear that f'(1/2) = 0, and in fact, f' is strictly decreasing on [0,1], so it can only be f'(x)=0 at one place (at x=1/2). You know f' is decreasing because:

f''(x) = - [ (x^2-1)ln(1-x) + x(2-2x+xln(x)) ] / x^2 (x+1)^2

You can quickly decide/verify for yourself that:



for all 0<x<1. This implies f'' is always negative, i.e. f' is always decreasing, i.e. f' can only be zero once on (0,1).

Because f(x) is differentiable on the interval (0,1) and continuous on [0,1], its extreme values can only occur at 0, 1, or places where f'(x)=0. This is only true when x=1/2.

f(0)=0 and f(1)=0, while f(1/2) is clearly not zero (it is ln(2)^2) so this is the unique maximum.

Advanced Math

All Answers

Answers by Expert:

Ask Experts


Clyde Oliver


I can answer all questions up to, and including, graduate level mathematics. I am more likely to prefer questions beyond the level of calculus. I can answer any questions, from basic elementary number theory like how to prove the first three digits of powers of 2 repeat (they do, with period 100, starting at 8), all the way to advanced mathematics like proving Egorov's theorem or finding phase transitions in random networks.


I am a PhD educated mathematician working in research at a major university.


Various research journals of mathematics. Various talks & presentations (some short, some long), about either interesting classical material or about research work.

BA mathematics & physics, PhD mathematics from a top 20 US school.

Awards and Honors
Various honors related to grades, various fellowships & scholarships, awards for contributions to mathematics and education at my schools, etc.

Past/Present Clients
In the past, and as my career progresses, I have worked and continue to work as an educator and mentor to students of varying age levels, skill levels, and educational levels.

©2017 All rights reserved.