Optimization
Use derivatives to find the best value of a quantity.
About this topic
Optimization uses derivatives to find the best value of something: the largest area, smallest cost, or maximum profit. Write a function for the quantity, using constraints to reduce it to one variable, find critical points by setting the derivative to zero, then check endpoints and confirm a maximum or minimum.
Example: a farmer has 200 m of fence for a rectangular pen against a long wall, so only three sides need fencing. With width x, length 200 − 2x, the area is A = x(200 − 2x) = 200x − 2x². Then A′ = 200 − 4x = 0 gives x = 50, so the pen is 50 by 100 and has area 5000 m². Since A″ = −4 is negative, this is a maximum.
Businesses minimise material for a can of fixed volume, airlines choose the most fuel-efficient speed, and nature itself finds optimal shapes, such as the hexagonal cells of a honeycomb.
Tips: draw a picture, define variables clearly, state the domain, and answer the question that was asked, such as the dimensions and not just the maximum area. A frequent error is skipping the constraint equation, which leaves two variables and no way to differentiate.
Key ideas
- Write the quantity to optimize
- Set the derivative to zero
- Check endpoints and the type of extremum
Quick check
Use cases: where Optimization helps
- Homework: review the key idea behind Optimization before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Optimization without notes.
- Visual learning: play with the interactive lab to see how Optimization changes when you adjust its inputs.
- Teaching a friend: explain Optimization in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Optimization?
Use derivatives to find the best value of a quantity.
What will I learn in Optimization?
Write the quantity to optimize. Set the derivative to zero. Check endpoints and the type of extremum.
Is Optimization part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Optimization lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Optimization?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.