# Where can I find example of Lagrange multipliers and constrained optimization?

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?

• I need any webpage address that contains these examples. – Huseyin Mar 18 '14 at 7:00
• Just curious, is this for a calculus class? I'm tutoring multivariable calculus and this could be of help. – Mark Fantini Mar 18 '14 at 10:54
• Yes @Fantini . I need more examples for Economics students in Calculus class (Math II). – Huseyin Mar 18 '14 at 15:16
• Ask the Economics faculty for relevant, simple to explain, examples? You might substitute "realistic" functions by simpler ones with roughly the right behaviour. – vonbrand Mar 19 '14 at 1:56
• @vonbrand I need exapmles. – Huseyin Mar 19 '14 at 6:45

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

• Second link was exactly which I had needed. – Huseyin Mar 19 '14 at 19:57