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For educational purposes, I am seeking example problems in the realm of natural sciences, engineering, and business that satisfy the following criteria:

  • Consider a one-dimensional real function $f$ (potentially with some parameters).
  • The problem should involve finding extrema of $f$.
  • It should be meaningful in this context to employ the first or second derivative test to confirm that the critical points identified are indeed extrema and to determine their nature (maximum/minimum). Importantly, this determination should not be evident through other means beforehand.
  • The calculations should be manageable manually, suitable for an advanced high school or freshmen level.
  • The problem should represent a realistic scenario in the specified field (natural sciences, engineering, or business), avoiding unrealistic constructions.
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    $\begingroup$ The best way I've found to collect problems of a certain type, or aspects of a certain topic, is described in this MSE answer. For what it's worth, many of my extensive Stack Exchange reference/application lists in the past 12 years (and in sci.math for at least 13 years before this) have been from sharing the results of some of these collections I've made. Example 1 & Example 2. $\endgroup$
    – Dave L Renfro
    Commented Mar 7 at 18:33
  • $\begingroup$ My favourite was to determine the optimal dimensions for a container or enclosure. $\endgroup$
    – Dan Christensen
    Commented Mar 11 at 14:05

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A Google search showed several that I bet won't satisfy you (fence enclosing, box making). I think you'd have to trawl through a lot of textbooks and look at end of chapter exercises and extract the few decent ones you find.

A classic example that is (I think) reasonably realistic is revenue maximizing for a monopolist (assume linear elasticity of demand, although you could have a higher function if you want). In addition to "working" and being reasonably real world (at least to being used in econ classes, not just math), I like that two competing factors are sort of intuitive and opposed. Higher price gets you more per purchase, but less purchases.

P.s. See similar question asked here, before (no killer compendium found): Optimization problems that today's students might actually encounter?

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FThe Problem: Given a specific volume, find the dimensions of a cylinder that minimize its surface area. This optimization problem is crucial in designing objects like Le Grand K, where the goal is to reduce air exposure and thereby slow the material's deterioration over time. For Le Grand K, a crucial design feature is that its height equals its diameter.

Why This Shape? Choosing a cylinder with height equal to its diameter optimizes the ratio of volume to surface area, a principle critical in the design of Le Grand K. This geometric optimization reduces the cylinder's exposure to air, minimizing the oxidation and wear over time, thus preserving its mass and, by extension, the standard of the kilogram.

Conclusion: This approach highlights the intersection of mathematical optimization and practical application in material science and metrology. By mathematically determining the dimensions that minimize surface area for a given volume, we can design objects like Le Grand K to ensure longevity and stability in physical standards, which are foundational to scientific measurement and research.

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