Could anyone give me a hint how to prove the following statements ? I suppose that I have some general ideas of pictures of functions satisfying the condition should look like. But it seems impossible for me now to give a prove.

  1. Suppose that $I \subseteq \mathbb{R}$ is an open interval and $f''(x)\geq 0$ for all $ x \in I$. If $c \in I$, show that the part of the graph of $f$ on $I$ is never below the tangent line to the graph at $(c,f(c))$.

  2. Let $f : [0,1] \rightarrow \mathbb{R} $ be continuous on $[0,1]$ and differentiable on $(0,1)$. Assume that $f(0)=0,f(1)=1$. Show that there are distinnt $c_1,c_2 \in (0,1)$ such that $f'(c_1)f'(c_2) = 1$.

  3. Give an example of a function $f : [0,1]\rightarrow \mathbb{R}$ such that $f$ is differentiable on $[0,1]$ but $\lim_{x \rightarrow 0} f'(x) $ does not exist.

  • $\begingroup$ I finally figure out how to prove 2. Any hints for 1 and 3 please. $\endgroup$ – Both Htob Sep 8 '14 at 12:01

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