I want give some extra homework about Lagrange multipliers and constrained optimization. I am interested in examples that are complex (for example, those having more than two variables and more than one restriction). Where I can find sufficient examples for students' practice?
Some simple examples you can try:
1) Maximize the product of $n$ positive numbers, given their sum is one.
2)Maximize entropy of $p_1,\dots, p_n$ (all $p_i>0$, summing to one) that is maximizing $-\sum p_i \log p_i$.
3) More conditions: Maximizing entropy, but with the extra condition that $ \sum a_i p_i = \mu$. You can also try introducing more conditions (variance?). The $a_i$s are known constants.
4) You can find a few more good problems here: https://www.math.ucdavis.edu/~thomases/W11_16C1_lec_2_4_11.pdf
5) A few more good problems here: http://homepages.math.uic.edu/~dcabrera/practice_exams/topics/lagrangemultipliers.html