- Topic
- Optimization
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Optimization practice question 5
Find the absolute maximum value of on the closed interval .
On a closed interval, the absolute extremes occur either at a critical point or at an endpoint. Both must be checked.
factors as .
Theory UBC Math 100 §8CLP-1 §3.5Key results
Worked solution
Strategy
Use the closed-interval method: find the critical points inside the interval, evaluate the function at those points and at both endpoints, and compare the list.
Working
Differentiate and factor. Both critical points lie inside [0, 5].
Evaluate at the endpoints. Skipping these is the usual way to lose this question.
Evaluate at the critical points.
The largest value is at the endpoint x = 5, well above the local maximum f(1) = 6.
Answer
The answer is 22, at the endpoint x = 5.
Check
f(x) − 22 = (x − 5)(x² − x + 4), and x² − x + 4 is always positive, so f(x) ≤ 22 for every x up to 5.
Where this goes wrong
Checking only the critical points and answering with the local maximum f(1) = 6, or answering f(3) = 2 by picking the wrong critical point. On a closed interval the endpoints are candidates too, and here x = 5 gives the largest value.
Leans on
Critical points, from Differentiation rules
Try a similar question
Find the local maximum value of .Saved in this browser only.