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function analysis, f(x)=ax has ≥1 solution for each a>0

Source: SEEMOUS 2008 P1

June 17, 2021
function

Problem Statement

Let f:[1,∞)→(0,∞)f:[1,\infty)\to(0,\infty) be a continuous function. Assume that for every a>0a>0, the equation f(x)=axf(x)=ax has at least one solution in the interval [1,∞)[1,\infty). (a) Prove that for every a>0a>0, the equation f(x)=axf(x)=ax has infinitely many solutions. (b) Give an example of a strictly increasing continuous function ff with these properties.
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