Optimization Strategy

Calculus 1 · Axiom Academy

A repeatable five-step method that turns any "biggest / smallest / cheapest" question into a derivative you can solve. Picture the quantity you care about plotted against the one variable you control. As that variable changes, the quantity rises, reaches a high point, then falls . At the very top the curve is momentarily flat — its slope is zero. That flat tangent is the whole idea: the maximum (or a minimum) hides wherever the derivative equals zero. Every optimization problem yields to the same five steps: Steps 1–2 set up Q=f(x) ; step 3 finds where it stops changing. 2. Walking the Strategy: Maximum Area The problem. You have 100 meters of fence to enclose a rectangular garden against a straight wall. The wall covers one side for free, so the fence only has to cover the other three. What dimensions give the largest area? Maximize A = xy . The fence gives x + 2y = 100 , so and . A'(x) = 50 - x . Solving A'(x)=0 gives x = 50 . , so the curve bends downward — x=50 is a maximum. Physically , and x=50 sits safely inside. Valid. Width x = 50 m and depth m give the maximum area . Notice the fence splits evenly — 50 m along the wall, 25 m on each side — the hallmark of an optimal rectangle against a wall. 3. The Step Everyone Forgets: the Domain

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