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Subsection 9.1.2 Consequences of Differentiability

There are a number of important consequences of a function being differentiable. These consequences are stated as mathematical theorems that you will need to know by name. We begin by introducing terminology about local extreme values.

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Definition 9.1.1 Local Maximum and Minimum

A function \(f\) has a local maximum at a point \(x=a\) if \(f(a) \ge f(x)\) for all \(x\) in a neighborhood of \(a\text{.}\) It has a local minimum at \(x=a\) if \(f(a) \le f(x)\) for all \(x\) in a neighborhood of \(a\text{.}\)

The first theorem is about the slope at a local extreme. It guarantees that a local extreme can only occur where the function either is not differentiable or has a horizontal tangent line.

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Theorem 9.1.2 Fermat's Theorem

If \(f\) has a local extreme at \(x=a\) and \(f'(a)\) exists, then \(f'(a)=0\text{.}\)

The second theorem combines the Extreme Value Theorem with Fermat's Theorem. If a function is continuous on a closed interval \([a,b]\text{,}\) then it must achieve both a maximum and a minimum value. If that function has \(f(a)=f(b)\text{,}\) then one of the extreme values must occur inside the interval at some point \(c \in (a,b)\text{.}\) If the function is also differentiable, then we must have \(f'(c)=0\text{.}\) This result is named Rolle's theorem.

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Theorem 9.1.3 Rolle's Theorem

If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\) and \(f(a)=f(b)\text{,}\) then there must be some value \(c \in (a,b)\) so that \(f'(c)=0\text{.}\)

The consequence of Rolle's theorem is that if a function starts and ends at the same value over an interval, it must turn around somewhere. For a differentiable function, the slope at that point must be \(f'(c)=0\text{.}\)

The third theorem about differentiability applies Rolle's theorem to say something about the average rate of change. Recall that the average rate of change,

\begin{equation*}
\left.\frac{\Delta f}{\Delta x}\right|_{[a,b]} = \frac{f(b)-f(a)}{b-a},
\end{equation*}

is the slope of the line, called a secant line, that joins the points \((a,f(a))\) and \((b,f(b))\text{.}\) The Mean Value Theorem guarantees that a continuous and differentiable function will have some point at which the tangent line has the same slope as the secant line over the interval.

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Theorem 9.1.4 Mean Value Theorem

If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\text{,}\) then there must be some value \(c \in (a,b)\) so that

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

Alternatively, we sometimes rewrite this as

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

###### Proof

Let \(s(x)\) be the linear function corresponding to this secant line and then define \(g(x)=f(x)-s(x)\text{.}\) Since \(s(a)=f(a)\) and \(s(b)=f(b)\text{,}\) we have \(g(a)=g(b)=0\text{.}\) If \(f\) is continuous and differentiable, then so is \(g\text{.}\) Rolle's theorem guarantees that \(g'(c)=f'(c)-s'(c) = 0\) for some value \(c \in (a,b)\text{.}\) Thus, \(f'(c)=\left.\frac{\Delta f}{\Delta x}\right|_{[a,b]}\text{.}\)