Optimization Worksheets
Use derivative signs and stated geometric constraints to interpret functions and rates.
Before you begin
Manipulate algebraic expressions and understand functions, limits and derivative notation.
Worked example
A rectangle has fixed perimeter 12 m and positive side lengths. What is the largest possible area?
- Let one side be x, so the other is 6 − x.
- A(x) = x(6 − x), whose derivative is 6 − 2x.
- The maximum occurs at x = 3, a square.
Answer: 9 m²
Check your result using the method in the example before moving to the next question.
Choose your level
These sets use the same exercise types and number ranges across level labels. Choose a fresh set for additional practice.
- Easy: Start with the preview and follow the worked example.
- Medium: Practice the same exercise types with a fresh numbered set.
- Hard: Use another set to practice explaining and checking your answers.
Questions and answer calculations checked on 2026-09-17. How we check worksheets. Free for personal and classroom use.
Preview and print an easy set
Try six questions below, or choose a full set at your level.
1
A rectangle has fixed perimeter 8 m and positive side lengths. What is the largest possible area?
2
A rectangle has fixed perimeter 20 m and positive side lengths. What is the largest possible area?
3
A rectangle has fixed perimeter 16 m and positive side lengths. What is the largest possible area?
4
A rectangle has fixed perimeter 24 m and positive side lengths. What is the largest possible area?
5
A rectangle has fixed perimeter 4 m and positive side lengths. What is the largest possible area?
6
A rectangle has fixed perimeter 12 m and positive side lengths. What is the largest possible area?
Answer Key
1. 4 m²2. 25 m²3. 16 m²4. 36 m²5. 1 m²6. 9 m²