Question

[Calculus 1 - Optimization Problems] Why do we disregard the radical when finding the minimum distance between two points on a graph? Details in description.

Answer 1

The square root function is monotonic increasing. Hence to find the minimizer of (f(x))^ 1/2 , it suffices to find the minimizer of f(x).

Answer 2

x<y iff sqrt x < sqrt y

Answer 3

Here’s another example. Suppose you want to find the minimizer of (f(x))^2 . Note that g(x)=x^2 is a strictly monotonic function with respect to the absolute value of x. Hence to find the minimizer of (f(x))^2 , it suffices to find the minimizer of |f(x)|.