Maximise the Absolute

Let mm and nn be distinct positive integers.

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

Note by Sharky Kesa
5 years, 7 months ago

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Simplest thing I can think of 1101|1^1-0^1|.... It has to be more complicated than this...

Trevor Arashiro - 5 years, 1 month ago

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

Sharky Kesa - 5 years, 1 month ago

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

Vivek Bhagat - 5 years, 1 month ago

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0

Asama Zaldy Jr. - 5 years, 1 month ago

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

Sparsh Goyal - 5 years, 1 month ago

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Note that it asks for a maximum value.

Sharky Kesa - 5 years, 1 month ago

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whats the answer

manish bhargao - 5 years, 1 month ago

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In terms of m and n?

Kenny Lau - 5 years, 1 month ago

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The answer should be "x" !

Sparsh Goyal - 5 years, 1 month ago

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Note that xrx^r is a strictly decreasing function when x(0,1)x\in(0,1), and the maximum value would be achieved when mm is the least and nn is the most, in this case, m=1m=1 and nn\to\infty, where xmxn=x10=x|x^m-x^n|=|x^1-0|=x.

Note: @Sparsh Goyal posted the numerical answer first.

Kenny Lau - 4 years, 11 months ago

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