Extrema are the high and low values of a function, and they help describe turning points, best choices, and limits of behavior. A local extremum is a highest or lowest value compared with nearby points, while an absolute extremum is the highest or lowest value on the entire domain being studied. In calculus, extrema connect graph shape to derivatives because many peaks and valleys occur where the slope is zero or undefined.
This distinction matters in optimization problems, graph analysis, physics, economics, and engineering.
Key Facts
- A local maximum at x = c means f(c) >= f(x) for x values near c.
- A local minimum at x = c means f(c) <= f(x) for x values near c.
- An absolute maximum on a domain means f(c) >= f(x) for every x in the domain.
- An absolute minimum on a domain means f(c) <= f(x) for every x in the domain.
- Critical points occur where f'(c) = 0 or f'(c) is undefined, as long as c is in the domain of f.
- Closed-interval method: find critical points in (a,b), evaluate f at those points and at endpoints a and b, then compare all values.
Vocabulary
- Absolute maximum
- The largest function value on the entire domain or interval being considered.
- Absolute minimum
- The smallest function value on the entire domain or interval being considered.
- Local extremum
- A maximum or minimum value compared only with function values at nearby inputs.
- Critical point
- An input inside the domain where the derivative is zero or does not exist.
- Closed-interval method
- A procedure for finding absolute extrema on a closed interval by checking critical points and endpoints.
Common Mistakes to Avoid
- Ignoring endpoints: endpoints can be absolute maxima or minima even if they are not critical points, so they must be evaluated on a closed interval.
- Assuming every critical point is an extremum: a derivative of zero can also occur at a flat point that is not a maximum or minimum.
- Confusing local and absolute extrema: a local maximum may be lower than another part of the graph, so it is not automatically the absolute maximum.
- Forgetting to compare function values: solving f'(x) = 0 only gives candidates, and the actual extrema come from evaluating f(x) at every candidate.
Practice Questions
- 1 For f(x) = x^2 - 4x + 1 on [0,5], find the absolute maximum and absolute minimum.
- 2 For f(x) = x^3 - 3x^2 + 2 on [-1,4], find all critical points, then determine the absolute extrema on the interval.
- 3 A continuous function on [a,b] has a local maximum at an interior point and its highest graph point occurs at the right endpoint b. Explain which point is the local maximum and which point is the absolute maximum.