How to solve
For a fixed perimeter the square maximizes area; for a fixed sum, equal halves maximize the product. Calculus confirms it: the optimum sits where the derivative is zero.
Steps
- Write the quantity as a function of one variable using the constraint.
- Set the derivative to zero.
- The optimum is the symmetric split — compute the requested value.
Example
A rectangle has perimeter 20. Largest possible area?
- Sides x and 10 − x; A = x(10 − x).
- A' = 10 − 2x = 0 → x = 5.
- A = 25.
Answer: 25
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