Hi guys, I'm not 100% sure what to do with the u (intensity of happiness) in this problem. Could someone give me a hint as to how I should go about this question?

mathmeme
May 9, 2017

#1**+3 **

Hi mathmeme

\(\mu\quad and \quad C \quad \text{are both constants}\\ Let \quad U(\theta) \text{ be referred to as U}\)

So we have:

\(U=\frac{\mu C}{\mu sin(\theta)+cos(\theta) }\\ U=\mu C(\mu sin(\theta)+cos(\theta) )^{-1}\\ \frac{dU}{d\theta}=-\mu C(\mu sin(\theta)+cos(\theta) )^{-2}(\mu cos(\theta)-sin(\theta))\\ \frac{dU}{d\theta}=\frac{-\mu C(\mu cos(\theta)-sin(\theta))}{(\mu sin(\theta)+cos(\theta) )^{2}}\\ \text{When }U'=0\\ -\mu C(\mu cos(\theta)-sin(\theta))=0\\ \mu cos(\theta)=sin(\theta)\\ \mu=tan(\theta)\)

Trouble is this could give a minimum or a maximum.

I did a graph in Desmos and found that mu = tan(theta) gives both the minimum and the maximum degree of happiness.

Here is the graph:

https://www.desmos.com/calculator/1c8zst7hlo

Feel free to question me about this answer :)

Melody
May 9, 2017