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

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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).
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x<y iff sqrt x < sqrt y
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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)|.
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