# 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
5 years, 4 months ago

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

- 4 years, 10 months ago

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

- 4 years, 10 months ago

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

- 4 years, 10 months ago

0

- 4 years, 10 months 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...

- 4 years, 10 months ago

Note that it asks for a maximum value.

- 4 years, 10 months ago

- 4 years, 10 months ago

In terms of m and n?

- 4 years, 10 months ago

The answer should be "x" !

- 4 years, 10 months ago

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.

- 4 years, 8 months ago

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