- Let $f(x)=|x|-1$. Then $f(-1)=f(1)=0$, but $f'(x) \neq 0$
on $ [-1,1] $. Does this contradict Rolle's Theorem? Explain!
- Does the Mean Value Theorem apply to the function $f(x)=\frac{x^2-4x
+3 }{ x-3}$ on $ [2,4] $?
- Is there a point $c$ on $ [2,4] $ for which $f'(c)= \frac{f(4)-f(2) }{
4-2}$ where $f(x)$ is the function of part $b)$?
Tags: rolle’s-theorem, MVT, absolute-value, illustrative-counterexample, multistep, using-a-theorem, a--v, a--a, t1, c2
In working through this problem, I find that students finally begin to understand that the MVT can only be applied when there is continuity on the closed interval and differentiability on the open interval.
– James Epperson 1998-08-09