## Problem 2

A company produces 300 mountain bikes weekly, which it sells entirely at a price of 600 euros each. After analyzing the market, it observes that if the price changes, sales also vary (continuously) according to the following proportion: for every 7 euros that the price of the bikes increases or decreases, sales decrease or increase by 3 units.

a) Can the company increase the price and obtain higher revenues?

b) At what price will the revenues be maximized?

## Solution to Problem 2:
At the current price, the weekly revenue is $600 \times 300 = 180,000$ euros.

(a) If the price is increased by 7 euros, then 297 bicycles will be sold, generating $607 \times 297 = 180,279$ euros in this case. So the answer to the first question is: Yes.

(b) Let $x$ be the amount in euros by which the price of each bicycle is increased ( $x$ can be any positive or negative real number, depending on whether the price increases or decreases), then the new price of each bicycle will be $600 + x$. According to the proportion in the statement, sales will vary to $300 - \frac{3}{7} x$ bicycles.

We consider the revenue function, where the variable is the price increase:

$$
f(x) = (600 + x) \times \left(300 - \frac{3}{7} x\right) = \frac{1}{7} \left(1,260,000 + 300 x - 3 x^2\right)
$$

The critical points of $f(x)$ are obtained by setting the derivative to zero: $f'(x) = (1 / 7)(300 - 6 x)$. The unique critical point is $x = 50$, which indeed gives a maximum of the function $f(x)$. Therefore, the solution to the question is: 650.