KindiakMath

An Increasing-Derivative Quotient

Problem 1. Let f : [0,\infty) \to \mathbb R be continuous and f(0) = 0. Suppose that f' exists and is increasing on (0,\infty). Prove that the function g(x) := f(x)/x is increasing on (0,\infty).

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Solution. It suffices to show that g' > 0 on (0,\infty). By the quotient rule,

\displaystyle g'(x) = \frac{xf'(x) - f(x)}{x^2}.

Since x^2 > 0 whenever x > 0, it suffices to prove that

\displaystyle f(x) < f'(x)x,\quad x > 0.

To that end, fix x > 0. By the mean value theorem, there exists c \in (0,x) such that

f(x) = f(0) + f'(c)(x-0) \quad \iff \quad f(x) = f'(c)x.

Since f' is increasing, c < x \Rightarrow f'(c) < f'(x). Hence,

f(x) = f'(c)x < f'(x) x,

as required.

—Joel Kindiak, 18 Oct 24, 1057H

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