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## Main Question or Discussion Point

Hi! I have the following question.

Let [itex]f[/itex] be continuous on [itex][0,\infty[[/itex], [itex]f(0)=0[/itex], [itex]f^\prime[/itex] exists on [itex]]0,\infty[[/itex], and [itex]f^\prime[/itex] is increasing on [itex]]0,\infty[[/itex].

the question is to prove that the following function is increasing that is [itex]g(x)=f(x)/x[/itex] on [itex]]0,\infty[[/itex].

I tried to show that the first derivative is positive but I did not succeed to use the monotonicity of [itex]]f^\prime[[/itex]

Let [itex]f[/itex] be continuous on [itex][0,\infty[[/itex], [itex]f(0)=0[/itex], [itex]f^\prime[/itex] exists on [itex]]0,\infty[[/itex], and [itex]f^\prime[/itex] is increasing on [itex]]0,\infty[[/itex].

the question is to prove that the following function is increasing that is [itex]g(x)=f(x)/x[/itex] on [itex]]0,\infty[[/itex].

I tried to show that the first derivative is positive but I did not succeed to use the monotonicity of [itex]]f^\prime[[/itex]