I am in charge of some practice lesson for Calculus II. I have to show how to apply the theory for unconstrained optimization (mainly Hessian analysis) and constrained optimization (Lagrange multipliers).

I'd like to use functions and boundaries taken from the real world. These examples must be easy and solvable and they should appear 'not-too-boring' to students.

Do you have any suggestion?

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    $\begingroup$ For "real life" example of undergraduate math topics (and high school math topics too) try COMAP comap.com $\endgroup$ Apr 20, 2019 at 14:59
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    $\begingroup$ By "real world", will you accept examples that are merely potentially real (involving believable scenarios and real objects), or do you want problems that someone got paid to solve? $\endgroup$
    – Nick C
    Apr 30, 2019 at 19:28
  • $\begingroup$ @Nick I'm interested in the first case.. Problems should be easy to solve for undergrad students: they have to be believable but not effectively realistic.. $\endgroup$ May 2, 2019 at 6:21
  • $\begingroup$ See also matheducators.stackexchange.com/questions/1550/… $\endgroup$ Aug 20, 2019 at 16:43

2 Answers 2


Typical constrained optimization (useful to "get it") is asking for the rectangle of largest area that can be enclosed in a fence of given length. Sure, it can be reduced to one-dimensional, but leave that option out. Or ask for the largest volume box with given surface area.


There are a few examples in Physics. Lagrangian Mechanics

  • $\begingroup$ True, but quite a bit too sophisticated for beginners. $\endgroup$
    – vonbrand
    Sep 25, 2019 at 18:56

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