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Section 9.1 Differentiable Functions

Subsection 9.1.1

Placeholder: Remember to address differentiability of rational powers somewhere.

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.

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.

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.

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.

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, \(f'(c)=\left.\frac{\Delta f}{\Delta x}\right|_{[a,b]}\text{.}\)