Question

You are given point (1,2).
f(x) = 0.199445983x^2 - 3.789473684x + 19
f'(x) = 0.398891966x - 3.789473684

You also know (0,19), (9.5,1) <-- this is the vertex.
This graph has the domain 0≤ x ≤ 9.5


Find the point on the curve that when joined to a line segment with this outside point (1,2), is the minimum distance that point could be away from the curve.

Answer 1

Refer to attached plot of your equations.

From the text of your problem and looking at the plot, I don't see where your line, colored red, has anything to do with a solution.

A line drawn through (1,2) at about 45 degrees with the x axis would intersect the curve near the point (4,7). A line drawn perpendicular to that line and passing through point (1,2) would meet the conditions stated in your problem. Of course I could be wrong.