Assertion: curvature's curvature | /r/CapableOf | reach local maximum
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text: From various comments and your edit to your post, my only remark about why the curvature is maximum at the point where the derivative = 0 is that if you imagine the curvature being plotted on a graph (a plot of curvature vs time $t$), there probably be a place where the curvature reaches some local maximum.