This came up in another thread.
We need to find the coordinates of M when BA is a maximum. Geogebra to the rescue!
1)The equation of the parabola can be determined by symmetry. It is:
2) Enter into the input bar f(x) = -2*x^2 - 4*x and press enter.
3) Enter into the input bar g(x) = x+2 and press enter.
4) Create and label 3 points O(0,0), C(-1,2), D(-2,0).
5) Pick a point that is on the left of C and place it on the parabola. Label it B.
6) Now drop a perpendicular line from B to the x axis.
7) Find the intersection point between this perpendicular line from B to the line x+2. Call this point of intersection A. ( use the intersection tool ).
8) Use the distance tool to measure the distance from B to A. It will appear in the algebra pane as distanceBA = something.
9) Set rounding to 10 digits.
10) use the move tool to slide point B along the parabola. See how close you can come to 1.125, I got 1.1249967. That is the maximum length of BA.
11) Go into the algebra pane and find x = -1.24941724. That is the x coordinate you seek.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.