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

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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
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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

Melody  May 9, 2017
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Thank you so much Melody! You're always so helpful :)

mathmeme  May 10, 2017
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Here's another graph to illustrate the point:

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

.

Alan  May 9, 2017
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Thank you for showing me the graph!

mathmeme  May 10, 2017