# Optimisation

Turn a real problem into a function of one variable, find its critical points, and pick the best one. Check the ends of the allowed range too.

Maths (university) · Part 04: Applications of derivatives · Card 4 of 9 · Last updated 2026-10-19

## Analogy

Finding the best seat in a theatre: define what 'best' means, then search for the peak.

## Example

**100 m fence, rectangular field**

Sides x and y: 2x + 2y = 100, so y = 50 - x.
Area A = x(50 - x) = 50x - x^2.
A' = 50 - 2x = 0 gives x = 25.
Square 25 by 25, area 625 m^2.

## In practice

Used to maximise area, profit or strength and to minimise cost, time or material.

## Practice

### Flashcards

- **Largest area of a rectangle with perimeter 100 m?** 625 m^2, from a 25 m by 25 m square.

### Interview questions

**A rectangle has perimeter 100 m. Find its dimensions for maximum area and justify that it is a maximum.**

- Let sides be x and 50 - x
- A = 50x - x^2
- A' = 50 - 2x = 0 gives x = 25
- A'' = -2 < 0 so a maximum
- Area is 625 m^2, a square

## Questions people ask

### What are the steps of an optimisation problem?

First, draw a picture and name the variables. Second, write the quantity to optimise as a function of one variable, using any constraint. Third, differentiate and solve for zero. Finally, check that it is a maximum or minimum and state the answer with units.

### Why must I check the endpoints?

The best value may sit at the edge of the allowed range, where the derivative is not zero. A function restricted to x between 0 and 10 might be largest at x = 10. Compare the critical points and the endpoints and choose the best.

## Related concepts

- [Second derivative test](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-3)
- [Critical point](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-1)
- [First derivative test](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-2)

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