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Suppose $f$ is continuous for $x\geq0$, $f'(x)$ exists for $x>0$, $f(0)=0$, and $f'$ is monotonically increasing. Then $g(x):=\cfrac{f(x)}{x}$ is monotonically increasing for $x>0$.Proof.Let $a *Note that the inverse doesn't hold.Example of $f$ such that the converse doesn't hold:

Excercise. (Principle of Mathematical Analysis 5.14) Let $f$ be a differentiable real function defined in $(a, b)$. Prove that $f$ is convex if and only if $f'$ is monotonically increasing. Answer.Recall: Definition of convex - $f$ is convex if and only if\[(1-\lambda)f(a)+\lambda f(b) \geq f((1-\lambda)a+\lambda b) \] for all $0\leq\lambda\leq1$ and $a, b$. WLOG, set $a\leq b$. Define \[L(x):=f..