Calculus Small Changes, Big Ideas

Section 23.2 More on the Mean Value Theorem

• Rolle’s theorem Let \(f:[a,b]\to\bb R\) be continuous on \([a,b]\) and differentiable on \((a,b)\text{.}\) If \(f(a)=f(b)\text{,}\) then there exists \(c\in(a,b)\) where \(f'(c)=0\text{.}\) (This can be used to prove the MVT.)

  • If a function has zero derivative on an interval, it must be constant.

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  • If two functions have the same derivative on an interval, they must differ by a constant.

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  • A function is monotone increasing if and only if its derivative is nonnegative on an interval.

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• Cauchy’s mean value theorem: If \(f\) and \(g\) are continuous on \([a,b]\) and differentiable on \((a,b)\text{,}\) then there is a number \(c\in (a,b)\) such that

  \begin{equation*} [f(b)-f(a)]\cdot g'(c)=[g(b)-g(a)]\cdot f'(c)\text{.} \end{equation*}

  Another way to visualize it is to have a parametric curve \((f(t),g(t))\) for \(a\le t\le b\) and point out that there exists \(c\in(a,b)\) such that

  \begin{equation*} \frac{f(b)-f(a)}{g(b)-g(a)}=\dfrac{f'(c)}{g'(c)} \end{equation*}

  (assuming that the denominators aren’t zero).

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• Cauchy’s MVT can be used to prove L’Hôpital’s rule.

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