פתרון בעיות אופטימיזציה

From setup to the winning number

• You've reduced the problem to one variable. Now solve the optimization.
• The plan: differentiate the objective, find critical points, and confirm which is the max or min.
• Then answer the actual question — often the dimensions or the optimal value, with units.
• It's the extremum machinery of this unit, applied to a story.

The solving steps

• 1. Differentiate the single-variable objective and set it to zero to find critical points.
• 2. Confirm max vs. min with the first or second derivative test.
• 3. Check the endpoints of the domain too, if the domain is closed or bounded.
• 4. Answer the question asked — plug the optimal variable back in.

Cut a square from each corner and fold: the volume peaks where its derivative is zero

Confirm, don't assume

• A critical point is only a candidate — you must show it's the max (or min) you want.
• Second Derivative Test: if $A''(x)<0$ at the critical point, it's a maximum.
• Or check the sign of $A'$ around it, or compare against domain endpoints.
• Skipping the confirmation is a common lost-mark; the AP expects a justification.

Answer the actual question

• If asked for the maximum area, report the area value; if asked for dimensions, report the lengths.
• Include units and check the answer is physically reasonable.
• Reread the prompt — "what dimensions?" and "what's the largest area?" want different final numbers.
• A number without the right label loses the point.