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This is a rather simple problem, but i can't for the life of me figure out the logic behind it.

The revenue $R $ from a software product depends on the price $p$ charged by the distributor according to the formula.

$$R = 4000p-10p^2$$

How sensitive is $R$ to $p$ when p is a) $100 $, b)$200$ c)$300 $?

Which begs for the differentiation: $\frac {dR}{dp}=4000-20p$

Now, finding the rate of change is easy. The rate of change is $2000$, $0$, $-2000$ for the respective values $100$, $200$, $300$.

If we were to maximize the revenue we would go for the $200$ option, because it is an absolute maximum value in the function $R = 4000p-10p^2$.

But practically, doesn't it mean that the revenue is $0$ dollars per dollar charged. And that a price at $200$ dollars would give us the revenue of $2000$ dollars per dollar charged?

Bullerskydd
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  • Note: I reformatted your post. The version of Latex used by this site uses "$$'s $" to indicate code intended for compiling, not as dollar signs. You can insert dollar signs, as I did in this comment, but it's usually more trouble than it is worth. – lulu Jan 01 '19 at 18:09
  • That said, I don't understand your question. The maximum revenue comes when the derivative vanishes (as there is clearly a global max and no global min). Thus at $p=200$, as you point out. But at $p=200$ we have $R(200)=4000\times 200 -10\times 200^2=400,000$ Where is the confusion? – lulu Jan 01 '19 at 18:12
  • The concept of rate of change is what set me up for confusion. I suppose it is synonymous with the slope at a given point. The slope at p = 100 is 2000. Doesn't this mean that for every dollar charged at this point, the income is 2000 dollars? Where as for p=200, where the slope is 0, there is 0 income for every dollar spent. Now I get that the maximum revenue is when the function reaches its peak. But how would we go about finding maximum revenue if there was no absolute maximum value at dR/dp=0? What if it would have been an inflection point? – Bullerskydd Jan 01 '19 at 18:25
  • Well, it's certainly true that this case is much simpler than the general case. Calculus isn't even needed...just complete the squares to write $R(p)=-10(p-200)^2+400000$ to make the global max visible. In general one does need to look at a second derivative, or nearby values, to test critical points. Not necessary here. – lulu Jan 01 '19 at 18:27

1 Answers1

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Plot the revenue curve and see how revenue changes at random $dp$ intervals on the price axis. At the maxima it barely changes with small price shifts, right? Since growth rate of parabola is very small at the start. Now check some remote intervals

Makina
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