I was given the following information about a function: Let $f$: [0,$\infty$) be a function so that
- $f(0)=0$
- $f$ is continuous on [0,$\infty$)
- $f$ is differentiable on (0,$\infty$)
- the function $f'$: [0,$\infty$) is monotonically increasing
Let $x, y \in (0, \infty)$ where $x < y$. We know that f is differentiable on (0,x) and (0,y) as well. By Mean Value Theorem there exists a $c \in (0, x)$ such that $f'(c) = \frac{f(x) - f(0)}{x}$=$\frac{f(x)}{x}$. And, there exists a $d \in (0, y)$ such that $f'(d) = \frac{f(y) - f(0)}{y}$=$\frac{f(y)}{y}$.
I was confused with the values $c,d$ I defined, Since $x<y$ does this mean that $c<d$? I have no idea if this is true but I want to know if theres any sort of relationship between $c$ and $d$ using this function $f$?