I was playing around in Desmos, and amidst that I became interested in how the graphs of exponentials and quadratics interact. For $0<a<1$ everything was as expected and nothing intriguing happened. However, for $a>1$ I noticed how $x^2=a^x$ had two real solutions for $1<a<2$. After playing around with the slider for a bit more I saw that this was true till about $a=2.1$ For any value of $a$ after that the graphs never met for $x>0$. So, I thought of finding the exact value of $a$ for which the two graphs just touch, or in other words, have exactly one solution.
At first I thought it would be trivial to find out at exactly what value of $a$ this happened. But when I actually sat down to try the problem, I realised I had no idea what to do. At first I thought the derivatives of the two functions would be equal when they touch, and even though that's true I found that the derivatives could be equal at other places too (not to say I have no idea how to solve $2x=a^x\ln a$ either)
I'm sure I am again making a big fuss out of something trivial. But, I just can't seem to see how we can solve this (with elementary maths). Can someone please point me in the right direction? Thanks
Best Answer
Why don't you write it as $\left(x^\frac{1}{x}\right)^2$ and draw its graph?
If I correctly remember, the maxima occurs at $x=e$ (easily verified by differentiating.)
So your value of $a$ for only one solution is $e^\frac{2}{e} \approx 2.087065$