Optimization on Closed Intervals

Real Analysis · Axiom Academy

Optimization on Closed Intervals A continuous function on a closed interval always has a best and a worst — and one recipe finds them: check the critical points and check the endpoints. A beverage company wants a can that holds exactly 355 mL using the least aluminum . The radius can't be anything you like — it lives between a too-skinny and a too-fat bound — so the search runs over a closed interval , where a best design is guaranteed to exist. Slide the radius, watch the cost Hold the volume fixed at 355 mL. A skinny can is tall and wastes side-wall; a fat can wastes lids. Somewhere between them the surface area bottoms out — drag and feel the curve dip. Three places to look — and only three New job: fence a rectangular pen with 100 m of fence , but a barn wall is one whole side, so fence only the other three. Maximize the area. Here's the only recipe you ever need. When the best price is in the middle Last one: you sell a gadget that costs 6 to make. Raise the price and each sale earns more but you sell fewer; drop it and you sell more but earn less. At what price does profit peak? Drag the price and let the recipe answer.

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