A house lies on a plain at point (3,4). There is a highway along the graph f(x) = x^3. I want to make the shortest path between the house and the highway. I need to find the point where the lane should meet the highway.
and a circle centred at (3,4) which touches the graph at a single point.
The radius of this circle is the minimum distance.
The point (x,y) where the circle meets the curve must satisfy
This is the distance of all points (x,y) on the curve from (3,4)
hence differentiating and setting the result to zero will find the x corresponding to minimum distance.
Here's another way to do it: the shortest line from a point to a curve must be perpendicular to the tangent line to the curve at the point where they cross.
The derivative of is so we must have a slope of for the perpendicular line that crosses the curve at . Since the line passes through (3, 4), it must be of the form . That line must also pass through so we must have
Yet another way- minimize the square of the distance subject to the constraint using Lagrange Multipliers.