Simple revision on differentiation

Problem 1. Suppose $f(1) = f(2) = 0$, $f(3) = 1$ and $f$ is twice differentiable on $[0,3]$. Show that $f''(c)>\frac{1}{2},\exists c\in(0,3)$.

Problem 2. Suppose $f(0) = 0, f(1) = 1$, $f$ is differentiable on $[0,1]$. Show that $\displaystyle \frac{1}{f'(a)}+\frac{1}{f'(b)}=2$, for some distinct $a,b\in (0,1)$.

兩條都是從某中學教師的 BLOG 中抽出來，後一條加上 distinct 的原因是為了把難度增加 (從 BLOG 中抽出來時沒有加上 distinct)。若把 distinct 移走，那很明顯存在 $f'(c)=1$, 取 $a=b=c$ 便完成證明。