Cue Column:
- How to set up optimization?
- What about constraints?
- How to verify max vs min?
Notes Column: Step-by-step method:
- Identify what to maximize/minimize (the objective function)
- Express it as a function of ONE variable using constraints
- Find the domain of the variable (physical constraints)
- Differentiate and find critical points (set f'(x) = 0)
- Verify using second derivative test or endpoint analysis
- Answer the question (may need to compute the max/min VALUE, not just the point)
Common constraint types:
- Perimeter fixed: P = 2(l+w), use to eliminate one variable from area A = lw
- Surface area fixed: eliminate one variable from volume
- Distance fixed: use Pythagorean theorem or distance formula
JEE Tip: If the domain is a closed interval, use the global extrema method (check endpoints too). If the domain is open or unbounded, use the second derivative test or limit analysis at boundaries.
AM-GM shortcut: For optimization of products with fixed sum (or sums with fixed product), AM >= GM often gives the answer faster than calculus. Example: For fixed a+b, ab is maximized when a = b.
Summary: Set up, reduce to one variable, differentiate, solve, verify.