How can we prove that $f_3 \colon (0, +\infty) \to \mathbb{R}$, $f_3(x) = x^{x^x}$ is monotonic? I tried taking the derivative — didn't work. Can we extend the proof to $f_{2k+1}$ for any $k$ (this function is constructed in a similar way, but $x$ is written $2k-1$ times. For example, $f_5(x) = x^{x^{x^{x^x}}}$)? All ideas are welcome.
Asked
Active
Viewed 137 times
1
-
1Do you mean "monotonic"? I suppose if you keep on taking powers of $x$ it may eventually become monotonous too... – Elchanan Solomon Mar 15 '22 at 10:22
-
Check this: https://math.stackexchange.com/q/1004017/42969 – Martin R Mar 15 '22 at 10:24
-
Or this: https://math.stackexchange.com/q/3279127/42969 – Martin R Mar 15 '22 at 10:28
-
Thanks, that helps a lot. Is there any way to extend the proof, though? – ABlack Mar 15 '22 at 10:41
-
Or this: https://math.stackexchange.com/q/365717/1508 – TonyK Mar 15 '22 at 12:08
-
@TonyK I will make sure to write it correctly from now on. It’s monoton in German so I just adapted the word I heard in lectures – ABlack Mar 15 '22 at 12:15