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Question
The total cost of producing x radio sets per day is Rs \[\left( \frac{x^2}{4} + 35x + 25 \right)\] and the price per set at which they may be sold is Rs. \[\left( 50 - \frac{x}{2} \right) .\] Find the daily output to maximum the total profit.
Solution
\[\text { Profit =S.P. - C.P}.\]
\[ \Rightarrow P = x\left( 50 - \frac{x}{2} \right) - \left( \frac{x^2}{4} + 35x + 25 \right)\]
\[ \Rightarrow P = 50x - \frac{x^2}{2} - \frac{x^2}{4} - 35x - 25\]
\[ \Rightarrow \frac{dP}{dx} = 50 - x - \frac{x}{2} - 35\]
\[\text { For maximum or minimum values of P, we must have }\]
\[\frac{dP}{dx} = 0\]
\[ \Rightarrow 15 - \frac{3x}{2} = 0\]
\[ \Rightarrow 15 = \frac{3x}{2}\]
\[ \Rightarrow x = \frac{30}{3}\]
\[ \Rightarrow x = 10\]
\[\text { Now,} \]
\[\frac{d^2 P}{d x^2} = \frac{- 3}{2} < 0\]
\[\text{ So, profit is maximum if daily output is 10 items.}\]
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