Suppose the profit $P(x)$ of a company is given by: $P(x) = 200x - 0.5x^2$, where $x$ is the number of units sold, find the rate of change of profit when $50$ units are sold?

150

The correct answer is Option (1) → 150 ##

Differentiate $P(x)$ with respect to $x$:

$\frac{dP}{dx} = \frac{d}{dx}(200x - 0.5x^2)$

$\frac{dP}{dx} = 200 – x$

At $x = 50$ units, the rate of change of profit is:

$\left. \frac{dP}{dx} \right|_{x=50} = 200 - 50 = 150$

The company's profit is increasing by $150$ units of currency when $50$ units are sold.