Additional Resources
OpenStax Online Textbook
How to Read a Math Textbook
Additional Instructional Resources
Maximum and Minimum Values- DA2
Objectives
- Define absolute extrema.
- Define local extrema.
- Explain how to find the critical points of a function over a closed interval.
- Describe how to use critical points to locate absolute extrema over a closed interval.
Summary
The critical numbers of a continuous function are the values of for which or does not exist. These values are important because they identify horizontal tangent lines or corner points on the graph, which are the only possible locations at which a local maximum or local minimum can occur.
Given a differentiable function whenever is positive, is increasing; whenever is negative, is decreasing. The first derivative test tells us that at any point where changes from increasing to decreasing, has a local maximum, while conversely at any point where changes from decreasing to increasing has a local minimum.
Given a twice differentiable function if we have a horizontal tangent line at and is nonzero, the sign of tells us the concavity of and hence whether has a maximum or minimum at In particular, if and then is concave down at and has a local maximum there, while if and then has a local minimum at If and then the second derivative does not tell us whether has a local extreme at or not.
To find relative extreme values of a function, we use a first derivative sign chart and classify all of the function's critical numbers. If instead we are interested in absolute extreme values, we first decide whether we are considering the entire domain of the function or a particular interval.
In the case of finding global extremes over the function's entire domain, we again use a first or second derivative sign chart. If we are working to find absolute extremes on a restricted interval, then we first identify all critical numbers of the function that lie in the interval.
For a continuous function on a closed, bounded interval, the only possible points at which absolute extreme values occur are the critical numbers and the endpoints. Thus, we simply evaluate the function at each endpoint and each critical number in the interval, and compare the results to decide which is largest (the absolute maximum) and which is smallest (the absolute minimum).