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