(Solved): Assume that the function f:[0,)R is continuous and that f(0)=0. In addition, assume that f is ...

Assume that the function $$f:[0,?)?\mathbb{R}$$ is continuous and that $$f(0)=0$$. In addition, assume that $$f$$ is differentiable on $$(0,?)$$ and that $$f^{?}$$ is strictly increasing, i.e., for all $$s,t?(0,?)$$ we have $$s<t?f^{?}(s)<f^{?}(t).$$ Use the Mean Value Theorem to prove that the function $$g:(0,?)?\mathbb{R}$$ defined by $$g(x)=f(x)/x$$ is strictly increasing.