One more :) Tough one on optimization. Thanks for helping out

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 :)

Here's another graph to illustrate the point:

.Seems odd that the "maximum" has a lower value than the "minimum"!!

.