A product costs the manufacturer ₹20 per unit. The demand function is given by $p(x) = 1000 - 20x$, then the quantity for maximum profit is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) - $\frac{49}{2}$ units
$\text{Cost per unit} = 20 \Rightarrow C(x) = 20x$
$\text{Demand function} \; p(x) = 1000 - 20x$
$\text{Revenue } R(x) = x \cdot p(x) = x(1000 - 20x)$
$= 1000x - 20x^2$
$\text{Profit } P(x) = R(x) - C(x)$
$= (1000x - 20x^2) - 20x$
$= 980x - 20x^2$
$\frac{dP}{dx} = 980 - 40x$
$980 - 40x = 0$
$x = \frac{980}{40} = 24.5$
The quantity for maximum profit is $24.5$ units.