Question

How to find points on parametric curve that attain minima?

Answer 1

I will assume that you know the formula for curvature at (x,f(x)), for a well-behaved curve y=f(x). It is
κ(x)=|y′′|(1+(y′)2)3/2.
In our case, the derivatives are easy to compute, and we arrive at
κ(x)=ex(1+e2x)3/2.
We wish to maximize κ(x). One can use the ordinary tools of calculus. It simplifies things a little to write t for ex. I imagine you can now complete the calculation. (The maximum curvature is not at x=0.)

Remark: One almost automatic reflex when we do the calculation is to maximize instead the square of the curvature. So we maximize e2x(1+e2x)3, or, more simply, t(1+t)3. With this version, it is difficult even for me to make a mistake