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I need some help on this equation:

$x\;$: Output Quantity
$CF\;$: Cost Function:
$CF(x)=5000+100x-\frac{x^2}{24}\;\leftrightarrow\;0\leq x\leq1600$

Now, I've already noticed that it sounds a litte bit strange, but, how can I calculate and find the maximum point of the output production quantity, instead of calculating the maximum marginal or total revenue in this situation? Please, show me how to approach and solve this problem.

Thank you for reading this and helping us all about...

PS: The problem asks for the maximum production point here.

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Sorry, question makes no sense. You ask for "the maximum point of the output production quantity," but you have never previously mentioned anything involving "production". You have mentioned "output quantity," but you can't be trying to maximize that, since $0\le x\le1600$ tells you that the maximum output quantity is 1600. Then you ask about the "maximum production point," but again it isn't clear what anything in the problem has to do with a "production point." A standard Calculus problem would be to maximize the TCF, but you haven't asked for that. Advice: post the question exactly... –  Gerry Myerson Mar 9 '12 at 12:25
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...as given to you. –  Gerry Myerson Mar 9 '12 at 12:25
    
Yes sir, I see your point here. Thank you for trying to taking care of this question kindly. first. Here, I've some idea now: C(x)=5000+100x-(x^2/24) , 0<=x<=1600; C`(x)=100-(x/12) (And, I tried to make the equation equals 0 after subtraction so, in that situation, x equals 1200. What about it now, does it make sense for the current solution approach?) –  Kerim Atasoy Mar 9 '12 at 12:40
    
@GerryMyerson: By the way, here is the exact question for your asking :D : " x mal miktarı olmak üzere toplam maliyet fonksiyonu: C(x)=5000+100x-(x^2/24) , 0<=x<=1600 'dür. Buna göre, maliyetin en yüksek (maksimum) olduğu üretim miktarı kaçtır? " It's been asked in the Turkish Language in a book which is used for studying for some exams... :) –  Kerim Atasoy Mar 9 '12 at 12:56
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up vote 3 down vote accepted

With $0\leq x\leq 1600$, the maximum value for the "output quantity" $x$ is $1600$.

At this point the "total cost" is falling, which seems a little strange.

As you say in a comment on the question, you can set the derivative of the cost function to $0$ to help trying to find a turning point in the cost function, and this will give you $x=1200$. Since the cost function is and upside down parabobla, the is the production point which maximises the cost (which is $65000$ at this point).

It turns out this is what you were asked for. Google Translate sends "olduğu üretim miktarı kaçtır?" to "What is the amount of production?" and "maliyetin en yüksek (maksimum)" to "cost is highest (maximum)"

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Yes sir, you might be right. I've solved 2 or 3 similar questions about this subjects using "Derivatives". They does require some knowledge, skills, formulas to solve. All I did is applying the C(x) and C`(x) formulas, then, making the equation equals 0. Does it sound logical for you? –  Kerim Atasoy Mar 9 '12 at 12:47
    
@Kerim Atasoy: What you are now saying makes sense as you are trying to find out what point of production maximises the cost. –  Henry Mar 9 '12 at 13:54
    
:D YES! :D I'm SOO~ proud of teaching you guys the Turkish Language indirectly by asking and learning about these problems... :D :) Thank you again... :) –  Kerim Atasoy Mar 9 '12 at 14:02
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