# 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
4 years ago

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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.

- 3 years, 4 months ago

The answer should be "x" !

- 3 years, 6 months ago

In terms of m and n?

- 3 years, 6 months ago

- 3 years, 6 months ago

0

- 3 years, 6 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...

- 3 years, 6 months ago

Note that it asks for a maximum value.

- 3 years, 6 months ago

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

- 3 years, 6 months ago

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

- 3 years, 6 months ago

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

- 3 years, 6 months ago