# Maximise the Absolute

Let $m$ and $n$ be distinct positive integers.

Find the maximum value of $|x^m - x^n|$, where $x$ is a real number in the interval $(0, 1)$.

Note by Sharky Kesa
6 years, 7 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

Note that $x^r$ is a strictly decreasing function when $x\in(0,1)$, and the maximum value would be achieved when $m$ is the least and $n$ is the most, in this case, $m=1$ and $n\to\infty$, where $|x^m-x^n|=|x^1-0|=x$.

Note: @Sparsh Goyal posted the numerical answer first.

- 5 years, 12 months ago

The answer should be "x" !

- 6 years, 1 month ago

In terms of m and n?

- 6 years, 1 month ago

- 6 years, 1 month ago

0

- 6 years, 1 month ago

I dont think it will have a numeric value as answer as "x" is variable and there are infinite nos. between 0 and 1...

- 6 years, 1 month ago

Note that it asks for a maximum value.

- 6 years, 1 month ago

Simplest thing I can think of $|1^1-0^1|$.... It has to be more complicated than this...

- 6 years, 1 month ago

He put (0,1), which means 0 & 1 not included, so it must be something else!!

- 6 years, 1 month ago

That doesn't work at all. I think you typoed.

- 6 years, 1 month ago