Exercise 13

Use the method of Lagrange multipliers to find the point on the line $x-2y=5$ that is closest to the point $(1,3)$. To do so, respond to the following prompts.

1. Write the function $f=f(x,y)$ that measures the square of the distance from $(x,y)$ to $(1,3)$. (The extrema of this function are the same as the extrema of the distance function, but $f(x,y)$ is simpler to work with.)

2. What is the constraint $g(x,y) = c$?

3. Write the equations resulting from $\nabla f = \lambda \nabla g$ and the constraint. Find all the points $(x,y)$ satisfying these equations.

4. Test all the points you found to determine the extrema.