Every optimisation problem has an objective, the thing being maximised or minimised, and a constraint restricting it.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Reduce to one variable
Solve the constraint for one variable and substitute into the objective. Only then can you differentiate.
Then the usual work
Differentiate, set to zero, and solve for the critical points.
Justification is required
The exam demands a reason: a sign chart for f prime, the second derivative test, or a closed-interval comparison.
Do not forget the ends
On a closed interval the extremum may sit at an endpoint, where the derivative is not zero.
Answer the question asked
If the question wants the maximum area, give the area rather than the x value that produced it.
The procedure
Write the quantity to optimise, use the constraint to reduce to one variable, differentiate, find critical points, and justify which is the extremum. The modelling is usually harder than the calculus.
The physical domain matters
Lengths cannot be negative and cannot exceed the material available. The endpoints of that domain must be checked alongside critical points, since the optimum sometimes sits at one.
Justify that it is the extremum
A sign chart for f′, or the second derivative test, or comparison of all candidate values. Reporting a critical point without showing it maximises rather than minimises is an incomplete answer.
Answer the question asked
If the question asks for the maximum area, the answer is the area, not the x that produced it. Substituting back and reporting the right quantity, with units, is the final step and it is routinely skipped.
Step 2: Try It Yourself
Tap and try it out.
- Point(2, 4)
- Slope of the tangent0
Step 3: Watch an Example
One step at a time.
Watch Rosa Maximise an Area
Rosa has 40 m of fence for a rectangular field and wants the greatest area.
- Step 1
The constraint is the perimeter: 2w + 2h = 40, so h = 20 − w.
Step 4: Your Turn
Practice makes it stick.
The Fence
Problem 1 of 2
With 40 m of fence, what is the largest rectangular area, in square metres?
m²
The Numbers
Problem 2 of 2
Two numbers add to 20. What is their greatest product?
Find the Best
1 of 8
With 24 m of fence, largest rectangular area?
2 of 8
Two numbers add to 10. Greatest product?
3 of 8
A = 20w − w². What is dA/dw at w = 4?
4 of 8
A = 20w − w². At which w is dA/dw zero?
5 of 8
Second derivative is negative at a critical point. Maximum or minimum? 1 max, 2 min.
6 of 8
Can the extremum sit at an endpoint of a closed interval? 1 yes, 0 no.
7 of 8
Put the optimisation method in order.
- 1Use the constraint to remove one variable.
- 2Differentiate and set the result to zero.
- 3Justify the extremum and answer the question asked.
- 4Identify the objective and the constraint.
8 of 8
Two numbers add to 30. Greatest product?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Two numbers add to 16. What is their greatest product?
Question 2 of 2
What does the exam require beyond finding the critical point?
What You Learned
- Pair an objective with a constraint and reduce to one variable.
- Differentiate, set to zero, then justify the extremum.
- Check endpoints and answer the question actually asked.