Optimisation of functions

Find two non-negative real numbers whose sum is exactly $20$, such that their product is as large as possible.

A mobile phone accessories company has determined that the monthly profit $B(x)$, expressed in euros, from selling $x$ custom phone cases is given by the function:

$B(x) = -x^2 + 120x - 500$

How many cases must be sold to achieve maximum profit? What is this maximum profit?

The visibility of a social media post (measured by the number of "likes") during the first $10$ hours after uploading follows the function:

$V(t) = 20t^2 - 2t^3$

where $t$ is the time in hours ($0 \le t \le 10$). At what moment does the post reach its peak visibility?

A group of students wants to fence off a rectangular area in the school playground for a charity market. If they have $40$ metres of fencing in total, determine the dimensions of the rectangle (length and width) enclosing the maximum possible area.

A rectangular wooden frame is to be made for an advertising poster that must have an area of $16$ $m^2$. Calculate the dimensions of the poster so that the cost of the wood for the frame is at a minimum (that is, so that the perimeter is at a minimum).

A young programmer is designing a time management application. He has determined that a user's productivity, measured in "focus points", follows the function $P(t) = -t^3 + 9t^2 + 21t$, where $t$ is the time in hours elapsed since the start of the working day ($0 \le t \le 8$). Calculate at what time of day the user reaches maximum productivity and how many points they obtain at that moment.

An open-topped cardboard box in the shape of a cuboid with a square base is to be designed for a new tablet model. The volume of the box must be exactly $2000 \text{ cm}^3$. Find the dimensions of the box (base side and height) that minimise the amount of cardboard used in its manufacture.

A delivery drone needs to travel from a charging point $A(0, 4)$ to a warehouse $B(10, 2)$. To optimize its battery, the drone must land at a point $P(x, 0)$ on a wireless charging pad located on the x-axis, and from there continue towards $B$.

The total energy cost function is given by the distance traveled: $E(x) = \sqrt{x^2 + 16} + \sqrt{(10-x)^2 + 4}$. Find the exact position of point $P$ that minimizes total energy consumption.

A technology company has launched a new social network. The number of active users, measured in millions, follows the function $U(t) = \frac{40t^2}{e^{0,5t}}$, where $t$ is the time in months since its launch ($t \geq 0$). Simultaneously, the monthly maintenance costs, in thousands of euros, are given by $C(t) = 10t + 5$. To match scales, we will work with $\frac{C(t)}{100}$ and define the difference $D(t) = U(t) - \frac{C(t)}{100}$.

1. a) Determine in which month the number of active users reaches its maximum level and calculate how many millions of users there are at that moment.

1. b) Calculate $D'(t)$ and study its sign using the calculator, trying for example $t = 3,9$ and $t = 4$. State between which two values the moment when the difference $D(t)$ is maximum lies and which month it approximately corresponds to.

1. c) Justify whether the maximum of $U(t)$ is absolute by analyzing the value of $U(0)$ and the long-term behavior of the function, that is, $\lim_{t\to+\infty} U(t)$.

The figure shows the graphs of $U(t)$ and of $\frac{C(t)}{100}$.

A market research study for a new personalised training application determines that the monthly profit $B(x)$, in hundreds of euros, depends on the subscription price $x$ (in euros), according to the function:

$B(x) = (x - 5)^2 \cdot \ln(x - 4)$ for $x > 4$.

Calculate the subscription price that maximises profit in the interval $[6, 15]$. Check whether there is any point of inflection in this interval indicating a change in the growth trend of profits.