When attempting to find the point on a curve that is closest to a fixed point
MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
When attempting to find the point on a curve that is closest to a fixed point, a student chooses to minimize the square of the distance rather than the distance itself. Which of the following best justifies that this choice yields the same location of the minimum?
(a) Because dxd[D2(x)]=2D(x)D′(x) differs from D′(x) only by the (non-zero) factor 2D(x), the two functions share the same critical points, so minimizing D2 minimizes D.
(b) Because the square function is strictly increasing on non-negative numbers, the point that minimizes D2 also minimizes D.
(c) Squaring removes the square root and produces a polynomial; therefore any critical point of D2 must automatically minimize the original distance D.
(d) Since the square function is one-to-one on all real numbers, the input that makes the square smallest must likewise make the original distance smallest.