Exercise 5.6

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Exercise 6: Suppose $f$ is continuous for $x\ge 0$, $f'(x)$ exists for $x>0$, $f(0)=0$, and $f'$ is monotonically increasing. Put $$g(x)=\frac{f(x)}{x}\quad(x>0)$$
    and prove that $g$ is monotonically increasing.

analambanomenos · Accepted Answer

By Theorem 5.10, for $x>0$ we have $f(x)=f(x)-f(0)=(x-0)f'(y)$ for some $y\in(0,x)$, so that $f(x)\le xf'(x)$ since $f'$ is monotonically increasing. Hence $$g'(x)=
    \frac{xf'(x)-f(x)}{x^2}\ge 0\quad(x>0),$$ and so $g$ is monotonically increasing by Theorem 5.11(a).

Exercise 5.6

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Exercise 5.6

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