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C20 · Unit C5 Function analysis and optimization

Optimization with a feasible domain

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 5.10–5.11
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

Translate an optimization problem into one objective and a feasible domain. Use a constraint to eliminate extra variables, find critical candidates, and justify a global conclusion using endpoints, sign changes or an inequality. A stationary value outside the physical domain is irrelevant. Include units and connect the result back to the original design.

Worked example

A rectangle with perimeter 40 m has sides x and 20 minus x, 0 is less than or equal to x is less than or equal to 20 if degenerate endpoints are included for comparison. A equals x(20 minus x), A minutes equals 20 minus 2x, giving x equals 10. Area 100 m squared exceeds endpoint areas 0, so the square is globally best.

Explore the mathematics

Optimization with a feasible domain

Connect a completed square to its vertex, roots and sign.

Setting 3 of 5

Scroll sideways to see the full diagram.

Predict the vertex and zeros of this shifted square.xy-4-3-2-11234y=(x−(0))²−1

Predict before continuing

Predict the vertex and zeros of this shifted square.

Read the complete explanation
  1. Predict the vertex and zeros of this shifted square.
  2. The square is smallest when x equals 0. The vertex has y=−1.
  3. Set y=0: (x−(0))²=1, so x=-1 or 1.
  4. The graph is below the axis between its roots and above it outside. Completing the square reveals the vertex.
See the idea: Optimization with a feasible domain
See the idea: Optimization with a feasible domainArea A=x(20−x) peaks at (10,100). The rectangle’s feasible side lengths run from 0 to 20; endpoints give zero area. Comparing endpoints and the stationary point proves the global maximum.xy005301060159020120maximum area 100

Area A=x(20−x) peaks at (10,100). The rectangle’s feasible side lengths run from 0 to 20; endpoints give zero area. Comparing endpoints and the stationary point proves the global maximum.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Finding A minutes equals 0 supplies a candidate; it is not a complete maximality argument. Preserve the physical constraint throughout.

Investigate and explain

Compare the optimum with a practical requirement such as integer cut lengths. Explain whether and why the solution changes.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.