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# $$k^x-x^k$$ Slider and $$x^y=y^x$$ graph

posted in: Algebra, Calculus, GeoGebra |

Use the slider to change the value of k, and to see dynamically how the graph of $$h(x)=k^x-x^k$$ changes. Negative values have $$x^k>k^x$$. Which value of k gives a graph that is never negative? Why?

Here is a very pretty graph of $$x^y=y^x$$ with areas where $$x^y < y^x$$ shaded in green and areas where $$x^y > y^x$$ shaded in purple.

I am working on an article about some problems related to these equations, including The Biggest Product Problem. There’s a lot of interesting stuff here!