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Real Analysis: Math 4050-4060

Valerio De Angelis

Section 5.2 Local extrema

Let \(I\) be an interval, \(f:I\longrightarrow {\mathbb R}\text{,}\) and \(c\in I\text{.}\)

Definition 5.2.1.

We say \((c,f(c))\) is a local maximum \(f\) if we can find some \(\delta \gt 0\) such that if \(x\in I\) and \(|x-c|\leq \delta\text{,}\) then \(f(x)\leq f(c)\text{.}\) In a similar way, a local minimum of \(f\) is a point \((c,f(c))\) for which we can find some \(\delta \gt 0\) such that if \(x\in I\) and \(|x-c|\leq \delta\text{,}\) then \(f(x)\geq f(c)\text{.}\) We say \((c,f(c))\) is a local extremum if it is either a local maximum or a local minimum.
Informally, \((c,f(c))\) is a local maximum means that \(f(c)\) is at least as high as any other value \(f(x)\) as long as \(x\) remains close enough to \(c\text{.}\) See the picture below. Note that the definition does not preclude the possibility that \(f(c)\) be equal to its neighbors. So for example if \(f(x)\) is constant, every point is a local maximum.
Figure 5.2.2. Local maximum
Figure 5.2.3. Local minimum
If the function is differentiable, it is clear from the geometric interpretation of the derivative as the slope of the tangent line that the derivative at a local extremum must be zero. In fact this is routinely used in the Calculus course to draw the graph of functions. The only other possibility is that \(f\) is not differentiable. The next theorem makes this precise.
Assume \(f\) is differentiable at \(x=c\) and consider the sign of the difference quotient \(\frac{f(x)-f(c)}{x-c}\) first as \(x\lt c\) and then as \(x\gt c\text{.}\)
In view of the last theorem, it is convenient to make the following definition.

Definition 5.2.5.

If \(f:I\longrightarrow {\mathbb R}\) and \(c\in I\) is not an endpoint of \(I\text{,}\) we say that \((c,f(c))\) if a critical point of \(f\) if either \(f'(c)=0\text{,}\) or \(f\) is not differentiable at \(x=c\text{.}\) The \(x\)-coordinate \(c\) of a critical point is called a critical number.
Note that by definition an endpoint of an interval \(I\) that is the domain of \(f\) is never a critical number. We can now rephrase Theorem 5.2.4 as: If \(c\) is not an endpoint of \(I\) and \((c,f(c))\) is a local extremum, then \((c,f(c))\) is a critical point.
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