The total cost of manufacturing x pocket radios per day is ₹ $\left(\dfrac{x^2}{4}+35x+25\right)$ and rate at which they may be sold to a distributor is ₹ $\left(\dfrac{100-x}{2}\right)$ each. What should be the daily output to attain a maximum total profit.
Daily output $=x$ radios
Cost price of x radios $=CP= ₹ \left(\dfrac{x^2}{4}+35x+25\right) $
Sale price of x radios $= SP= ₹ \; x \left(50 - \dfrac{x}{2}\right) = ₹ \left(50x - \dfrac{x^2}{2}\right)$
$\therefore$ Profit function $P\left(x\right) = SP - CP= \left(50x - \dfrac{x^2}{2} - \dfrac{x^2}{4} - 35x - 25\right)$
i.e. $P\left(x\right) = 15x - \dfrac{3x^2}{4}-25$
$\therefore$ For maximum profit, $\dfrac{dP}{dx} = 0$
Now, $\dfrac{dP}{dx} = 15 - \dfrac{6x}{4}$
$\therefore$ $\dfrac{dP}{dx}=0$ $\implies$ $15 - \dfrac{3}{2}x = 0$
i.e. $\dfrac{3}{2}x = 15$ $\implies$ $x = 10$
$\therefore$ Daily output $=10$ radios.