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now the profit, p(x) = revenue(r(x)) - cost for manufacture (c(x)) is a universal truth. If it's negative means it's just a lost and not profit.

The profit should be maximum when p'(x) = 0. As can be seen:
enter image description here
but, what if the function for r(x) and c(x) were such, even though it's highly impossible in real life the function exists;
enter image description here

Would this universal relationship not hold? What could the possible flaw in the logic be?

Community
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mathmaniage
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1 Answers1

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I don't know whether I understand your question correctly. $p(x)=r(x)-c(x)$. You make a profit if your revenue is more than the cost as you mentioned. Now If you have to maximize profit, first you need to find the critical point by letting $p'(x)=0$. In other words critical point is the point at which marginal cost $c'(x)$ equals marginal revenue $r'(x)$. If for this $x$, if you find that $r''(x)<c''(x)$, only then your profit will be maximized. In short, the "universal relation" will always hold.

creative
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  • I am confused on one point though. Profit is maximum when c(x) is minimum and r(x) is maximum, In other words the difference in r(x) and c(x) must be the highest . This is what r'(x) = c'(x) gives. In the second graph I have illustrated this. The difference is maximum yet the derivative is not equal. – mathmaniage Apr 06 '17 at 13:55
  • above you have mentioned that r''(x) < c"(x) still would only imply that the revenue is decreasing in relation with the cost, which is directly given by the graph itself. – mathmaniage Apr 06 '17 at 13:58
  • As you mentioned "the difference in $r(x)$ and $c(x)$ must be the highest".. This happens only if $r''(x)<c''(x)$ which in turn (for max profit) must happen for that $x$ for which $p'(x)=0$. Actually I studied all these in my $+2$. I just shared with you whatever I could recollect. – creative Apr 06 '17 at 14:04
  • well, I don't know, what you say about r''(x) < c''(x) is true but r'(x) being equal to c'(x) isn't being logically sensible from the second illustration above – mathmaniage Apr 06 '17 at 15:02