Nick's Curve Question

A photo of a whiteboard titled: Nick’s Curve Question

My colleague Nick Cheng asked about the curve $f(p) = x^p - x$ for $x \in (0,1)$ and $p \in (0,1)$. He was interested in where the maximum is for various values of $p$.

We check the end-points and see: $f(0) = f(1) = 0$. These are probably not our maxima. Now, we look for critical points: \[ f'(p) = px^{p-1} - 1 = 0 \Longrightarrow x^{p-1} = \frac{1}{p} \Longrightarrow x = p^{-\frac{1}{p-1}} \] And so, a bit of calculus says that the max should occur at $x = p^{-\frac{1}{p-1}}$.

You can play with Nick’s Curve on Desmos.
The locus of possible maxima is highlighted in this Desmos.

Nick's Curve Question

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