- 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.

**Theorem 2. (Fermat’s Theorem)**: Suppose \(f\) is a function that is defined on an open interval containing the point \(c\). If \(f(c)\) is a local maximum or minimum, then either \(f\) is not differentiable at \(c\) (meaning \(f'(c)\) does not exist) or \(f'(c)=0\).

- 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.

#### Show the proof

#### Hide the proof

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 \(f(x)≥f(x_0).\) ↗