There is a webcam in my office that I use to take photos of the whiteboard. This setup was inspired by Dror Bar Natan’s Blackboard Shots. You can read more about the setup here. If you know of anyone else with a similar setup, please let me know.
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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 $x = p^{-\frac{1}{p-1}}$.
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