Determine the minimal distance from point (-3,3) to the curve given by y= (x -3)^3

Question

Determine the minimal distance from point (-3,3) to the curve given by y= (x -3)³

Answer 1

We will minimize distance ^ 2 (no reason to deal with square roots) and call this D

The distance^2 from (-3, 3) to (x, f(x)) = (x, (x-3)^3) is

(x - -3)^2 +( (x-3)^3 - 3)^2 =

x^2 + 6x + 9 + (x-3)^6 - 6(x-3)^3 + 9

DD/dx = 6(x-3)^5 - 18(x-3)^2 + 2x + 6

This equals 0 when x = 2.27811834518233. The distance is √((x - -3)^2 +( (x-3)^3 - 3)^2) = 6.265552