Q8Multiple correct4 Marks24 Dec 2023

An LED manufacturer determines that in order to sell $x$ number of LEDs, the price per LED (in thousands) must be $f(x) = 1000 - x$, if $x \le 800$, and the manufacturer also determines that the total cost (in thousands) of producing $x$ number of LEDs is $$g(x) = \begin{cases} 30000 + 300x, & \text{if } x \le 400 \\ 100x + 110000, & \text{if } 400 < x \le 800 \end{cases}$$ Although in the above context, $x$ can take only integer values, assume that $x$ is a continuous variable in the interval $[0, 800]$ and that the functions $f(x)$ and $g(x)$ are defined as above on this entire interval. Let $P(x)$ denote the function representing the profit of the company. Choose the set of correct statements.

A

P(x) is continuous in the interval [0, 800].

B

The function P(x) has two local maxima in the interval [0, 800].

C

A global maxima of P(x) lies in the interval [500, 800].

D

All global maxima of P(x) lies in the interval [300, 700].