- In many practical problems, we need to find the greatest (maximum) value or the least (minimum) value—there can be more than one of each—of a function.
- The maximum and minimum values of a function are called the extreme values or extrema of the function.
- Extremum is the singular form of extrema. The plural forms of maximum and minimum are maxima and minima, respectively.
- Differentiation can help us locate the extreme values of a function.
In calculus, there are two types of “maximum” and “minimum,” which are distinguished by the two prefixes: absolute and relative.
Table of Contents
Absolute Maxima and Minima
The concepts of absolute maximum and minimum were introduced in Chapter 4. Let’s review the definitions.
Let the function \(f\) be defined on a set \(E\). We say \(f\) has an absolute maximum on \(E\) at a point \(p\) if \[f(x)\leq f(p)\quad\text{for all }x\text{ in }E,\] and an absolute minimum value on \(E\) at \(q\) if
\[f(x)\geq f(q)\quad\text{for all }x\text{ in }E.\]
Absolute maxima and absolute minima (plural forms of maximum and minimum) are also referred to as global maxima and global minima.
Previously, we learned that:
Theorem 1. The Extreme Value Theorem: If \(f\) is continuous on a closed interval \([a,b]\), then \(f\) attains both its absolute maximum \(M\) and absolute minimum \(m\) in \([a,b]\). That is, there are numbers \(p\) and \(q\) in \([a,b]\) such that \(f(p)=M\) and \(f(q)=m\).
We should emphasize that:
- The continuity of the function on an open interval (instead of a closed interval) is not sufficient to guarantee the existence of the absolute maximum and minimum of the function.
- If the function fails to be continuous even at one point in the interval \([a,b]\), the extreme value theorem may fail to be true (although a discontinuous function may have max and min).
For more information, see the Section on the Extreme Value Theorem.
Local (or Relative) Maxima and Minima
- Geometrically speaking local maxima and local minima are respectively the “peaks” and “valleys” of the curve.
Definition 1. A function \(f\) is said to have a local (or relative) maximum at a point \(c\) within its domain \(D\) if there is some open interval \(I\) containing \(c\) such that \[f(x)\leq f(c)\quad\text{for all }x\in{I}.\] The concept of local (or relative) minimum is similarly defined by reversing the inequality.
- Every absolute maximum or minimum that is not an endpoint of an interval is a local maximum or local minimum, respectively. An endpoint is precluded from being a local extremum because we cannot find an open interval around an endpoint that is contained in the domain of the function.
- Notice that differentiability, or even continuity, of \(f\) at other points is not required.
- The geometrical interpretation of the above theorem is: At a local max or min, \(f\) either has no tangent, or $f$ has a horizontal tangent.
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We shall give the proof for the case of a local minimum at \(x=c\). According to the definition, we have \[f(c)\leq f(c+h)\] or \[0\leq f(c+h)-f(c)\] for all \(h\) sufficiently close to zero (that is, when \(c+h\) is near \(c\)). If \(f'(c)\) does not exist, there is nothing else to prove. So suppose \[f'(c)=\lim_{h\to0}\frac{f(c+h)-f(c)}{h}\] exists as a definite number. We need to show \(f'(c)=0\). When \(h\) is small, we have \[\frac{f(c+h)-f(c)}{h}\geq0\quad\text{if }h>0\] and \[\frac{f(c+h)-f(c)}{h}\leq0\quad\text{if }h<0\] because the numerator in both cases is either positive or zero (\(f(c+h)-f(c)\geq0\)). If we let \(h\to0^{+}\), from the first case, we have
\[f'(c)\geq0,\] and if we let \(h\to0^{-}\), from the second case, we have \[f'(c)\leq0.\] Because we have assumed that \(f'(c)\) exists, we must have the same limit in both cases, so \[0\leq f'(c)\leq0.\] This can happen only when \(f'(c)=0\). The proof for the case of a local maximum is similar.
The above theorem states a necessary condition for a local extremum. That the condition is not sufficient is evident from a glance at the point \((r,f(r))\) in Figure 9. The graph of \(f\) has a horizontal tangent at this point, but \(f\) does not have an extreme value at \(x=r\). As another example, consider: \(f(x)=x^{3}\)
\[f(x)=x^{3}\Rightarrow f'(x)=3x^{2}\]
\[f'(0)=0\] but \(x=0\) does not give either a local maximum or a local minimum of \(f\), as is obvious from the graph of \(y=x^{3}\) (Figure 10(a)). If \(g(x)=\sqrt[3]{x}\), then
\[g(x)=x^{1/3}\Rightarrow g'(x)=\frac{1}{3}x^{1/3-1}=\frac{1}{3}x^{-2/3}=\frac{1}{3\sqrt[3]{x^{2}}}\]
and \(g'(0)\) is not defined (we may say \(g'(0)=+\infty\)), but \(g(0)=0\) is not a local extremum (Figure 10(b)).
Critical Points
A number in the domain of the function at which the derivative is zero or the derivative does not exist has a special name. It is called a critical number.
Definition 2. Critical point: A point \(c\) in the domain of a function \(f\) is called a critical point (or critical number) of \(f\) if \[f'(c)=0\quad\text{or}\quad f'(c)\text{ does not exist.}\]
The number \(f(c)\) is called a critical value of \(f\).
- Recall that if \(f'(c)=+\infty\) or \(f'(c)=-\infty\), we say \(f'(c)\) does not exist because \(+\infty\) and \(-\infty\) are not numbers.
By the above definition, we can reword Fermat’s theorem as:
Fermat’s Theorem: If \(f(c)\) is a local maximum or a local minimum, then \(x=c\) is a critical number of \(f\).
- According to the above theorem, every single local extreme value is a critical value, but not every critical value is necessarily a local extreme value.
- We mentioned that every absolute extreme value, with the exception of an absolute extreme value that occurs at an endpoint, is also a local extreme value. Hence:
An absolute maximum or minimum of a function occurs either at a critical point or at an endpoint of its domain.
This provides us a method to find the absolute maximum and the absolute minimum of a differentiable function on a finite closed interval \([a,b]\).
Strategy for finding the absolute extrema of a continuous function \(f\) on a finite closed interval \([a,b]\):
- Step 1: Find \(f'(x)\)
- Step 2: Find all critical values: Set \(f'(x)=0\) and solve it for \(x\). Also find every value of \(x\) for which \(f'(x)\) does not exist. Evaluate \(f\) at each of these numbers that lie between \(a\) and \(b\).
- Step 3: Evaluate \(f(a)\) and \(f(b)\).
- Step 4: The largest value of \(f\) from Steps 2 and 3 is the absolute maximum of \(f\) and the least value of \(f\) from these steps is the absolute minimum of \(f\) on \([a,b]\).
1\(x≤y\) means \(x<y\) or \(x=y.\), so we can write, for example, \(2≤2.\) Here \(f(x)=f(x_0)\) for all \(x\) in \(I\), and therefore we can write \(f(x)≤f(x_0)\) or \(f(x)≥f(x_0).\) ↗