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I need to maximise $P$ under these conditions. First, $$P=N[1.4W-0.31L]\,\text{ and }\,W\le19,\,7\le{L}\le11,\,N\in\{16,17\}.$$ I did this using elemantary methods. In order to maximise $P$, we need to maximise $[1.4W-0.31L]$, and I thought since $W$'s coefficient is positive and $L$'s coefficient is negative, to maximise we need to use maximum $W$ and minimum $L$, so $W=19$ and $L=7$, then we get $24,43$. Since we are looking for $\max\{P\}$, we need maximum $N$, and $N$ can be either $16$ or $17$ so $\max\{P\}=415,31$.

I have solved this question like above. However, my teacher asked me to write an essay and I reached this question and I need to do higher level mathematics like use of derivative. Can you suggest me another solution which involves more complicated mathematics?

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ciceksiz kakarot
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  • It looks like what you're doing is just linear programming, and you've done it correctly... no derivatives necessary... In general, it's not good to look for more complicated mathmatics - simpler and more elegant is often better. – Zen May 30 '13 at 17:58
  • @Zen I have to write the essay and one of the criteria is usage of mathematics and I do not have any mathematics other than this kind of things, that is why I need to find another way. You said linear programming,do you know how we draw graphs when we have intervals, I know a little linear programming and that is when there is all functions(no interval), I drew the functions and find the intersection part. How do I do it with intervals such as 7≤L≤11 – ciceksiz kakarot May 30 '13 at 18:04

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Your solution is fine. Yes, you can write things like $\frac {dP}{dW}=1.4N$ and argue that since this is never zero the maximum must occur at one end of the interval, and since $N \gt 0$ it must be the high end, so $W=19$. Really this is just showing off the fact that you can write the derivative and using the reasoning you already have. I would clarify with your teacher whether the point of the essay is to expand the reasoning you give or to describe the process of using the derivative so that if you had a non-linear function you would find the optimum. If the second, it would be better to pick a function that has its optimum in the range instead of at one end (or do one of each.)

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    If you really wanted to show off you could write $\frac{\partial P}{\partial W}$ since P is a function in three variables ;) – Zen May 30 '13 at 18:08
  • I do not think my teacher would accept partial derivates since the essay should be on high school level – ciceksiz kakarot May 30 '13 at 18:25