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The Powers that Be
As you probably know, 2 and 4 are the only integer values for which the following equality holds: xy=yx. (The value is equal to 16.) [NB. given x≠y.]
But what about the rest of the function?
Mathematica to the rescue:
ContourPlot[(x^y) - (y^x), {x, 0, 4}, {y, 0.00001, 4},
AspectRatio -> Automatic];
UPDATE: Aha! Here we go. I've refined the plot display so that you can see the function better. It's not quite as seductive, but a little more to-the-point. Of course the diagonal line represents the case where x=y.
But what about the rest of the function?
Mathematica to the rescue:
ContourPlot[(x^y) - (y^x), {x, 0, 4}, {y, 0.00001, 4},
AspectRatio -> Automatic];
UPDATE: Aha! Here we go. I've refined the plot display so that you can see the function better. It's not quite as seductive, but a little more to-the-point. Of course the diagonal line represents the case where x=y.