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In \[y = 48x - 2{x^2}\],
Where $ y = $ total revenue in \[ \$ \] ,
 $ x = output $ ,
At what output is the total revenue a maximum?
A. $ 2 $
B. $ 12 $
C. $ 48 $
D. $ 4 $

Last updated date: 13th Jul 2024
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Hint: The question is asking us to find the maximum value of the total revenue denoted in this case by $ y $ , which corresponds to the value a company makes in dollar based on the value of output produced which here is denoted by $ x $ , Thus we can say that $ y $ is the function of $ x $ . And we have to find the maxima of the function. When we find the maxima of the function we first put the derivative of the function equal to $ 0 $ and then find the critical points, check the values of those points at the second derivative and if that value is negative the function is maxima at that critical point. The given function can be easily differentiated by the use of a standard formula.

Complete step-by-step answer:
We have to find the value of the maximum of the given function for revenue against the output. We will first differentiate this function. The differentiation is ,
 $ \Rightarrow y = 48x - 2{x^2} \\
   \Rightarrow y' = 48 - 4x \;
Upon putting $ y' $ as $ 0 $ we will get the value of critical points,
 $ \Rightarrow 0 = 48 - 4x $ ,
 The critical point is therefore $ 12 $ . This is the value of the output when the revenue of the company is maximum. The option B is therefore correct.
So, the correct answer is “Option B”.

Note: We could have checked these critical points further in case we got more than one critical point but since we got only one critical point we can safely assume it will be for a maxima, because options do not have any none of these options. This is a very important lesson to avoid unnecessary calculations when the answer is obvious, but remember to do it only when multiple choice questions are given, do not do it in the subjective type question to avoid unnecessary deduction of marks.