#### 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

\[\text { Profit =S.P. - C.P}.\]

\[ \Rightarrow P = x\left( 50 - \frac{x}{2} \right) - \left( \frac{x^2}{4} + 35x + 25 \right)\]

\[ \Rightarrow P = 50x - \frac{x^2}{2} - \frac{x^2}{4} - 35x - 25\]

\[ \Rightarrow \frac{dP}{dx} = 50 - x - \frac{x}{2} - 35\]

\[\text { For maximum or minimum values of P, we must have }\]

\[\frac{dP}{dx} = 0\]

\[ \Rightarrow 15 - \frac{3x}{2} = 0\]

\[ \Rightarrow 15 = \frac{3x}{2}\]

\[ \Rightarrow x = \frac{30}{3}\]

\[ \Rightarrow x = 10\]

\[\text { Now,} \]

\[\frac{d^2 P}{d x^2} = \frac{- 3}{2} < 0\]

\[\text{ So, profit is maximum if daily output is 10 items.}\]