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The Mean Value Theorem is one of the most important theorems in calculus. We look at some of its implications at the end of this section. First, let’s start with a special case of the Mean Value Theorem, called Rolle’s theorem.
Informally, Rolle’s theorem states that if the outputs of a differentiable function $f$ are equal at the endpoints of an interval, then there must be an interior point $c$ where $f\prime \left(c\right)=0.$ [link] illustrates this theorem.
Let $f$ be a continuous function over the closed interval $[a,b]$ and differentiable over the open interval $\left(a,b\right)$ such that $f\left(a\right)=f\left(b\right).$ There then exists at least one $c\in \left(a,b\right)$ such that $f\prime \left(c\right)=0.$
Let $k=f(a)=f(b).$ We consider three cases:
Case 1: If $f\left(x\right)=0$ for all $x\in \left(a,b\right),$ then $f\prime \left(x\right)=0$ for all $x\in \left(a,b\right).$
Case 2: Since $f$ is a continuous function over the closed, bounded interval $[a,b],$ by the extreme value theorem, it has an absolute maximum. Also, since there is a point $x\in \left(a,b\right)$ such that $f\left(x\right)>k,$ the absolute maximum is greater than $k.$ Therefore, the absolute maximum does not occur at either endpoint. As a result, the absolute maximum must occur at an interior point $c\in \left(a,b\right).$ Because $f$ has a maximum at an interior point $c,$ and $f$ is differentiable at $c,$ by Fermat’s theorem, $f\prime \left(c\right)=0.$
Case 3: The case when there exists a point $x\in \left(a,b\right)$ such that $f\left(x\right)<k$ is analogous to case 2, with maximum replaced by minimum.
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An important point about Rolle’s theorem is that the differentiability of the function $f$ is critical. If $f$ is not differentiable, even at a single point, the result may not hold. For example, the function $f\left(x\right)=\left|x\right|-1$ is continuous over $[\mathrm{-1},1]$ and $f\left(\mathrm{-1}\right)=0=f\left(1\right),$ but $f\prime \left(c\right)\ne 0$ for any $c\in (\mathrm{-1},1)$ as shown in the following figure.
Let’s now consider functions that satisfy the conditions of Rolle’s theorem and calculate explicitly the points $c$ where $f\prime (c)=0.$
For each of the following functions, verify that the function satisfies the criteria stated in Rolle’s theorem and find all values $c$ in the given interval where $f\prime \left(c\right)=0.$
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