Section 6.3 Maxima and minima
In this section we study how to find maxima and minima of functions. We distinguish between local and absolute extrema of a function. We study the Extreme Value Theorem, which provides a simple approach to finding the absolute max and min of a function on a closed interval. Overall, the content of this section is very useful to sketch the graph of complicated functions (Section 6.4), but also to solve optimization problems, which are very common in science, economics, etc. (Section 7.2)
Objectives
You should be able to:
- Explain and illustrate the definition of local and absolute extrema.
- Explain and illustrate the definition of critical numbers of a function.
- Relate critical numbers of a function to its local extrema.
- Find the critical numbers of a function.
- Explain and illustrate geometrically the Extreme Value Theorem.
- Find the absolute minimum and maximum of a function on a closed interval.
Subsection 6.3.1 Instructional video
Subsection 6.3.2 Key concepts
Concept 6.3.1. Min and max of a function.
Let \(c \in D\) where \(D\) is the domain of \(f\text{.}\) Then \(f(c)\) is:
- The absolute maximum of \(f\) on \(D\) if \(f(c) \geq f(x)\) for all \(x \in D\text{;}\)
- The absolute minimum of \(f\) on \(D\) if \(f(c) \leq f(x)\) for all \(x \in D\text{;}\)
- The local maximum of \(f\) on \(D\) if \(f(c) \geq f(x)\) for all \(x\) near \(c\text{;}\)
- The local minimum of \(f\) on \(D\) if \(f(c) \leq f(x)\) for all \(x\) near \(c\text{.}\)
Near \(c\) means “for all \(x\) in some open interval containing \(c\text{.}\)”
In general, absolute minima and maxima can occur either at local minima or maxima, or, if \(D\) is a closed interval, at the endpoints of the interval.
Concept 6.3.2. Extrema and critical numbers.
A critical number of \(f\) is a number \(c\) in the domain \(D\) of \(f\) such that either \(f'(c) = 0\) or \(f'(c)\) does not exist.
If \(f\) has a local minimum or maximum at \(c\text{,}\) then \(c\) is a critical number of \(f\) (Fermat's Theorem). However, the converse is not true: not all critical numbers are local minima or maxima.
Concept 6.3.3. Extreme Value Theorem.
If \(f\) is continuous on \([a,b]\text{,}\) then \(f\) must attain an absolute maximum \(f(c)\) and an absolute minimum \(f(d)\) at some numbers \(c,d \in [a,b]\text{.}\)
Concept 6.3.4. How to find the absolute extrema of a continuous function \(f\) on a closed interval \([a,b]\).
- Find the critical numbers of \(f\) in \((a,b)\text{,}\) and evaluate \(f\) at the critical numbers;
- Evaluate \(f\) at the endpoints \(a\) and \(b\text{;}\)
- Compare the values. The largest of these values is the absolute maximum, while the smallest is the absolute minimum.