The mathematics is rarely the hard part. Naming the variables and writing the objective is where most applied problems are won or lost.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Use the constraint early
A constraint often lets you eliminate a variable outright, turning a two-variable problem into a one-variable one.
Closed regions
On a closed bounded region, an absolute maximum and minimum are guaranteed to exist. They must be somewhere.
Check the boundary
The extremes may sit on the edge rather than at an interior critical point. Checking only the interior misses them.
Corners too
On a rectangular region the corners are separate candidates and must be evaluated individually.
Check the answer makes sense
A negative length or a fractional person means the model, not the arithmetic, has gone wrong.
The modelling is the hard part
Writing the objective and the constraint from a described situation is where these problems are won or lost. The calculus afterwards is mechanical by comparison.
Constraint or substitution
A constraint can sometimes be solved and substituted, reducing to an unconstrained problem in fewer variables. When it cannot be solved cleanly, Lagrange multipliers are the alternative.
Physical restrictions bound the domain
Lengths and quantities cannot be negative, and materials are limited. Those restrictions define the region, and the optimum may sit on its boundary rather than at an interior critical point.
Verify and interpret
Confirm which candidate is the extremum sought and state the answer in the terms the question used, with units. A critical point reported without that is an incomplete answer.
Step 2: Try It Yourself
Tap and try it out.
Nested closed rings around one point mean a single peak or valley.
Step 3: Watch an Example
One step at a time.
Watch Amina Design a Box
Amina needs an open-topped box with a square base and volume 32 cubic centimetres, using the least material.
- Step 1
She names the base side x and the height h, so the volume gives x²h = 32.
Step 4: Your Turn
Practice makes it stick.
The Box
Problem 1 of 2
A box has a square base of side 4 and volume 32. What is its height?
The Two Numbers
Problem 2 of 2
Two positive numbers sum to 30 and their product is largest. What is that product?
Model and Solve
1 of 8
Two numbers sum to 20 with the largest product. What is that product?
2 of 8
On a closed bounded region, is an absolute maximum guaranteed? 1 yes, 0 no.
3 of 8
A rectangular region has how many corners to check separately?
4 of 8
An open box with square base side 2 and volume 12. What is its height?
5 of 8
Two numbers sum to 14 with the largest product. What is that product?
6 of 8
A model gives a side length of −3 metres. Is that a valid answer? 1 yes, 0 no.
7 of 8
Sort each location by whether it must be checked when hunting absolute extrema on a closed region.
Tap something to move it.
- Empty
- Empty
8 of 8
A square base of side 5 with height 4. What is the volume?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Two numbers sum to 24 with the largest product. What is that product?
Question 2 of 2
Why must the boundary be checked on a closed region?
What You Learned
- Naming the variables and writing the objective is the hard half of an applied problem.
- A constraint usually lets you eliminate a variable before differentiating.
- On a closed region, check interior critical points, the boundary and the corners.