Originally Posted by

**lingyai** I’ve just finished my textbook, “Forgotten Calculus”. The last chapter is on multivariable calculus. Disappointingly, while there were some “pure math” problems on finding and classifying relative extrema, there were no real-world applications of this technique, i.e optimization problems.

So I decided to make one up, or rather base it on a marginal profit problem in the text. So what I’ve done below is not an “assigned” question per se. So the book has no answer for me to check myself by.

I would therefore ask any of you kindly, mentoring types out there to tell me if I am going about this right or wrong (I suspect the latter) and if need be correct me. To make it easier I show pretty much all of my work, although it does make this post pretty long.

I also need some guidance on interpretation.

Lastly, if anyone can recommend any sources which have well-explained, worked examples in multivariate optimization, I’d be grateful. I don't think I need (or have time for) a whole book on the subject; 5-10 examples would help a lot.

**The premise**

Total annual profit z from a small “summer hobby” farm is determined as

z = f(x,y) = 10x^3 + 20y^2 – 10xy

when x = weekly ad spending = 10 and

y = acres farmed = 5.

This much came from the text.

**The question and strategy summary**

I’d like to find out which combination(s), if any, of ad spending and acres maximizes (or minimizes) profit.