Question

A(1,4),B(6,1))find point c on x-axis so AC+BC is a minimum

Answer 1

What an interesting problem! If the sum of the two distances AC and BC is to be minimized, we could set up an expression L = |AC| + |BC| in terms of unknown x-coordinate c, then use differential calculus in the usual manner to determine critical value(s) and test which one leads to a minimum L.

Another way, probably not acceptable in math classes where fancy calculators are not allowed, would be to set up and graph function L. One should be able to estimate visually the value of c at which the graph reaches "rock bottom".

Hint: The distance between two points (a,b) and (c,d) is Sqrt( (c-a)^2 + (d-b)^2).