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Problem 33 Hard Difficulty

If $ p(x) $ is the total value of the production when there are $ x $ workers in a plant, then the $ average productivity $ of the workforce at the plant is
$ A(x) = \frac {p(x)}{x} $
(a) Find $ A'(x). $ Why does the company want to hire more workers if $ A'(x) > 0? $
(b) Show that $ A'(x) > 0 $ if $ p'(x) $ is greater than the average productivity.

Answer

A. $A(x)$ is increasing
B. $A^{\prime}(x)>0$

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Video Transcript

here we have an equation which stands for the average productivity, and we want to find its derivative. So we're going to use the quotient rule. So we have the bottom times, the derivative of the top, minus the top times, the derivative of the bottom over the bottom squared, and we can go ahead and simplify that just a little bit. We don't really need a times one. So we have X Times p prime of X minus P of X over X squared. All right, so imagine that that is positive. If it's positive, why would the company want to hire more workers? So this is the rate of change of average productivity. And if the rate of change is positive, it means productivity is increasing. As Ex gets bigger, the average productivity would increase. So we we would think the company would want their productivity to increase. So more workers Woodley to ah, higher productivity. If it was the other way around and if the derivative was negative, more workers would actually lead to less productivity, so that would not be ideal. Okay, Now, for Part B, we're going to show that the rate of change is greater than zero. If we're showing that if the rate of change of productivity is greater than the average productivity, then the rate of change of the average will be positive. Okay, that's a mouthful. So we start with this. The rate of change of productivity is greater than the average. We want to show that this leads us to the rate of change of the average is greater than zero. What are the steps in between? Well, let's go ahead and multiply both sides by X. And we can do that in this inequality because we know that X is positive. X stands for the number of workers and we wouldn't have a negative number of workers. So let's go ahead and do that, and we get X Times P Prime of X is greater than P of X. Now we could subtract p of X from both sides, and we have X times p prime of X minus p of X is greater than zero. And do you notice how this is looking more and more like the rate of change we found in part A. The only thing it's missing is it doesn't have it divided by X squared in it. So what this inequality still be true if we were to divide it by X squared X squared is positive. If you divide by a positive, you're not going to change the inequality at all. So XP prime of X minus p of X over X squared is greater than zero. So that is saying that the rate of change of average productivity is greater than zero.