5.11 Solving Optimization Problems

The steps

1) Identify the objective and the constraint. 2) Use the constraint to write the objective in one variable. 3) Set the derivative equal to 0 to find critical points. 4) Verify it's a max or min (First/Second Derivative Test or endpoints).

Finish carefully

Consider the domain of feasible values, and state the answer with the actual quantity and units.

Key terms

Optimization steps

Objective + constraint → one variable → critical points → verify.

Find critical points

Set the derivative to zero.

Verify

Use the First or Second Derivative Test (or endpoints).

Domain

Restrict to feasible values.

Answer

State the real quantity with units.

Try an AP-style question

Equal squares of side x are cut from the corners of a 12 × 12 sheet to make an open box. Which x maximizes the volume?

1. 3
2. 2
3. 6
4. 4