- In many practical problems, we need to find the greatest (maximum) value or the least (minimum) value—can be more than one of each—of a function.
- The maximum and minimum values of a function are called
**extreme values**or**extrema**of the function. - Extremum is the singular form of extrema. The plural form of maximum and minimum are maxima and minima.
- 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.

### 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**.

We learned that:

**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 Section 4.9.

### Local (or relative) maxima and minima

Figure 1 shows the graph of function \(f\) with absolute maximum at \(x=b\) and absolute minimum at \(x=p\). The point \((c,f(c))\) is higher than all nearby points on the curve, although it is not as high as \((b,f(b))\). That is, if we consider only values of \(x\) sufficiently close to \(c\), then \(f(c)\) is the largest value of those values of \(f\) (Figure 2). In this case, we say \(f\) has a local (or relative) maximum at \(x=c\).

Similarly, we say \(f\) has a local (or relative) minimum at \(x=d\) because \(f(x)\geq f(d)\) for all \(x\) very close to \(d\) (Figure 1). Other local maxima and minima are denoted on Figure 3.

- Geometrically speaking local maxima and local minima are respectively the “peaks” and “valleys” of the curve.

The precise definitions are as follows.

*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 \[f(x)\leq f(c)\quad\text{for all }x\in\ensuremath{I}.\] 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 and 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.

Because \(x_{2}\not\in Dom(f)\) , the function cannot have a maximum there. Moreover, it is evident that \(f(x_{4})\) is a local maximum. We claim that \(f(x_{6})\) is a local maximum because if we zoom in, we realize that for all \(x\) close enough to \(x_{6}\), \[f(x)\leq f(x_{6}).\]

All the absolute and local extrema are shown in Figure 6.

It is evident from Figure 9 that at a local extremum, the tangent line is either parallel to the \(x\)-axis (slope = 0) or has no tangent line. The following theorem helps us locate all the possible values of \(c\) for which there is a local extremum.

**Theorem. (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

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 7. 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. 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 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 \([a,b]\).

- \(x\leq y\) means \(x<y\)
**or**\(x=y\). So we can write \(2\leq2\). Here \(f(x)=f(x_{0})\) for all \(x\) in \(I\), and therefore we can write \(f(x)\leq f(x_{0}\)) or \(f(x)\geq f(x_{0}).\)↩︎