Finding the minimum distance between two curves

[Charles Link]

I got x1=1.8317... x2=2.5421...

Excellent. By @Gavran's method, I have $x_2=\frac{a}{5}+\frac{12}{5} \approx 2.54$. :) (see also posts 24, 25, 29, and 30).

Note: In the OP and the first several posts following it, we solved for the minimum distance between $y=4-x^2$ and $y=(x-3)^2$. In these last couple of posts we are solving for the minimum distance between $y=4-x^2$ and $y=(2x-6)^2$.

 

[Charles Link]

Find the two points that share the same normal line. It would be a general way as @pasmith showed.

Yes, this way seems to get the result a little easier. I worked the $y=4-x^2$ and $y=(2x-6)^2$ problem by this method just now, and it does seem somewhat easier than other methods. i.e. Find the two points with same slopes that share the same normal line.

@anuttarasammyak I see your additions to post 30 including a couple graphs. Excellent. :)