Calc 1 Optimization Question

Question

Find the point on the graph of the function that is closest to the given point.

f(x) = (x-1)^2

Point: (-5,3)

Answer 1

The distance d between any point (x,(x−1)²) and (-5, 3) is given by the distance formula:

d=sqrt((x+5)²+((x−1)²−3)²)

We can minimize the square of this distance function since that is the same as minimizing the distance function itself.

D(x) = (x+5)²+((x−1)²−3)²

Take the first derivative and set it equal to zero to find critical points:

D'(x) = 2x³ − 6x² + x + 9 = 0

Using a graphing calculator, numerical methods (like Newton's Method), or an online solver, the real root of this equation is x = -1.

We can confirm this is a local minimum since D''(-1) is > 0.

At x = -1, f(-1) = 4

So, the point closest to (-5, 3) is (-1, 4)

Hope this was helpful.