The closed-interval method is a reliable way to find the absolute maximum and absolute minimum of a continuous function on a closed interval [a,b]. It matters because many real problems ask for the largest or smallest possible value, such as maximum height, minimum cost, or greatest profit within allowed limits. The key idea is that on a closed interval, the highest and lowest values must occur either at an endpoint or at a critical point inside the interval.
This method turns an optimization problem into a short checklist of values to compare.
To use the method, first confirm that the function is continuous on the entire interval. Then find critical numbers in the open interval (a,b), where f'(x) = 0 or f'(x) does not exist. Evaluate the original function f(x), not the derivative, at every critical number and at both endpoints a and b.
The largest output is the absolute maximum value, and the smallest output is the absolute minimum value.
Key Facts
- Closed-interval method applies when f is continuous on [a,b].
- Extreme Value Theorem: a continuous function on [a,b] has both an absolute maximum and an absolute minimum.
- Critical numbers occur where f'(x) = 0 or f'(x) is undefined, as long as x is in (a,b).
- Candidate points are x = a, x = b, and all critical numbers in (a,b).
- Evaluate f(x) at every candidate point, then compare the output values.
- Example: if f(x) = x^3 - 3x on [-2,2], then f'(x) = 3x^2 - 3, critical numbers are x = -1 and x = 1.
Vocabulary
- Closed interval
- A set of x-values from a to b that includes both endpoints, written [a,b].
- Absolute maximum
- The greatest value of a function over a specified domain or interval.
- Absolute minimum
- The smallest value of a function over a specified domain or interval.
- Critical number
- A number c in the domain of f where f'(c) = 0 or f'(c) does not exist.
- Continuous function
- A function whose graph has no breaks, holes, or jumps on the interval being considered.
Common Mistakes to Avoid
- Forgetting to check the endpoints is wrong because absolute extrema on a closed interval often occur at x = a or x = b.
- Using f'(x) values instead of f(x) values is wrong because the maximum and minimum are determined by the function outputs, not the derivative outputs.
- Including critical numbers outside the interval is wrong because only points in the given closed interval can be candidates for absolute extrema.
- Assuming f'(x) = 0 finds every candidate is wrong because critical numbers can also occur where the derivative does not exist.
Practice Questions
- 1 Use the closed-interval method to find the absolute maximum and minimum of f(x) = x^2 - 4x + 1 on [0,5].
- 2 Use the closed-interval method to find the absolute maximum and minimum of f(x) = x^3 - 6x^2 + 9x + 2 on [0,5].
- 3 Explain why the endpoints must be checked when finding absolute extrema on a closed interval, even if the derivative has critical numbers inside the interval.