The following exercise is again about the Mean Value Theorem :)
Let $f : [0,1] \rightarrow \mathbb{R}$ be continuous and differentiable on $(0,1)$. Assume that $ \lim_{x\rightarrow 0^+} f'(x)= \lambda.$
Show that $f$ is differentiable (from the right) at $0$ and that $f'(0)=\lambda$.
Hint: Mean Value Theorem.
What exactly is 'differentiable from the right'? Do I have to show that the limit as $x$ approaches $0$ from the right side exists? How can I do that?
How can I use the Mean Value Theorem to show that the derivative on $0$ equals $\lambda$?
Mean Value Theorem: If $f:[a,b] \rightarrow \mathbb{R}$ is continuous on $ [a,b] $ and differentiable on $ (a,b)$, then there exists a point $ c \in (a,b)$ where
$ f'(c)= \frac{f(b)-f(a)}{b-a}.$