Step by Step
W
Write the objective function
Identify and write the quantity you're trying to maximize or minimize — this is your objective function.
I
Identify the constraint
Find the equation that relates your variables due to some given restriction (like a fixed perimeter or fixed budget) — this is your constraint equation.
R
Reduce to one variable
Use the constraint equation to solve for one variable in terms of the other, then substitute it into the objective function, leaving you with a function of a single variable.
E/D
Extremize, then Decide max or min
Take the derivative of this single-variable objective function, set it equal to zero to find critical points, then use the second derivative test or compare endpoint values to determine whether each critical point is actually a maximum or a minimum.
Applied Walkthrough
1
A farmer has 100 feet of fencing and wants to enclose the maximum possible rectangular area against a barn wall (no fence needed on that side). Write the objective: Area = x·y.
2
Identify the constraint: only two widths and one length need fencing (since the barn wall covers the fourth side), so 2x + y = 100.
3
Reduce to one variable: solve the constraint for y (y = 100 − 2x) and substitute into the objective: Area = x(100 − 2x) = 100x − 2x².
4
Extremize: take the derivative, dA/dx = 100 − 4x, set it to 0, giving x = 25. Using the second derivative test (d²A/dx² = −4, which is negative, confirming a maximum), this is indeed the maximum. Plugging back in, y = 50, giving a maximum area of 1,250 square feet.
Exam Application
Exams test whether you can correctly reduce a two-variable optimization problem to a single variable using the constraint, and whether you verify — rather than assume — that a critical point is actually a maximum or minimum.
⚠ Common Trap
The most common trap is finding a critical point and assuming it's automatically the answer, without verifying whether it's a maximum or a minimum using the second derivative test or by comparing endpoint values.
✓ Quick Self-Check
1. What does the WIRED mnemonic stand for?
Write objective, Identify constraint, Reduce to one variable, Extremize, Decide max or min.
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2. What is the role of the constraint equation in an optimization problem?
It lets you reduce the objective function from two variables down to one, by solving for one variable in terms of the other.
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3. How do you verify whether a critical point is a maximum or minimum?
Using the second derivative test or by comparing values at the endpoints of the domain.
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4. In the fencing example, what was the maximum enclosed area?
1,250 square feet, when x = 25 and y = 50.
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5. What is the most common mistake in optimization problems?
Finding a critical point and assuming it's the answer without verifying it's actually a maximum or minimum.
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