Saturday, March 27, 2010
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。
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。
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