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 Inverse Crossed Linear Power (Posted on 2023-10-14)
Determine all possible real x that satisfy this equation:

(1/x)x = 4x+1/16

 See The Solution Submitted by K Sengupta Rating: 5.0000 (1 votes)

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 another solution? | Comment 4 of 13 |
I'm not sure whether there is another actual solution but the LHS (1/x)^x is defined for negative rational numbers with an odd denominator.

The RHS is always positive, so the rational solution would also have an even numerator.

By graphing |x|^(-x) you can get a better idea of what's going on.  The crosses the RHS according to Desmos at around -0.3261

https://www.desmos.com/calculator/prvzcru2ut

A pretty good approximation of this decimal is -14/43 which you can see is pretty close to a solution.

If the actual solution is irrational, we can get arbitrarily close to it in this manner.  I doubt there is an exact rational form as for the positive soltuons.

 Posted by Jer on 2023-10-15 12:24:30

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