Exercises: Function optimisation

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

A cell phone accessories company has determined that the monthly profit $B(x)$, expressed in pesos, 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?

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

It is required to split the number $12$ into two positive parts such that the sum of their squares is a minimum. Use derivatives to find these parts.

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 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).

The manufacturing cost of a batch of wireless headphones is given by the function $C(x) = 2x^2 - 80x + 2000$, where $x$ is the number of units produced. How many units must be manufactured to minimise total production cost?

A farmer knows that if he plants $100$ orange trees on his plot, the average yield per tree is $400$ kg of fruit per year. For each additional tree planted, the average yield per tree decreases by $2$ kg due to overcrowding. The total yield is given by:

$P(x) = (100 + x)(400 - 2x)$

where $x$ is the number of additional trees planted. Find the value of $x$ that maximises the total annual yield.

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.

A professional cooking workshop designs an open-topped steel container with a square base for preserving food. The container must have a volume of $13.5$ liters ($13.5\ \text{dm}^3$). Determine the dimensions of the base side ($x$) and height ($h$) that minimize the amount of steel used (that is, the total surface area of the container) and calculate this minimum surface area.

The total manufacturing cost of $x$ units of a wireless headphone model is given by the function $C(x) = 4x^2 + 200x + 10\,000$ pesos. The unit selling price depends on demand and follows the formula $p(x) = 1\,400 - 2x$. Determine how many units must be manufactured and sold for the company's total profit to be at a maximum.

A marketing study for a streaming platform indicates that the monthly number of subscribers, in thousands, varies according to the monthly subscription price $x$ (in pesos) following the function $S(x) = \frac{48}{x} + x - 10$ for prices between $4$ and $15$ pesos. Calculate the subscription price that minimizes the number of subscribers. Justify whether this result is useful for the company from the revenue perspective.

An open-topped cardboard box in the shape of a rectangular prism 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 minimize the amount of cardboard used in its manufacture.

A "booktuber" has found that the impact of their videos (measured in thousands of daily views) depends on the time $t$ (in days) elapsed since publication, according to the function $V(t) = \frac{100t}{t^2 + 9}$. Determine on which day the video reaches its maximum impact and what that maximum number of views is.

An advertising poster for a music festival is to be printed with a total area of $2 \text{ m}^2$. The poster must have margins of $10 \text{ cm}$ at the top and bottom, and $5 \text{ cm}$ at the left and right sides. Calculate the total dimensions of the poster so that the area of the printed region is as large as possible.

An online urban clothing store has estimated that its daily revenue in pesos follows the function $I(p) = -2p^2 + 120p$, where $p$ is the average price in pesos of its garments. On the other hand, its daily expenses are modeled by $G(p) = 40p + 100$. Find the price $p$ they should set so that the daily profit (Revenue $-$ Expenses) is maximum. Also determine the amount of this maximum profit.

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 pesos are given by the function $C(t) = 10t + 5$.

Determine in which month the number of active users reaches its maximum level. At what time is the difference between the number of users (in millions) and the maintenance cost (divided by 100 to align scales) at its maximum? Justify whether this maximum is absolute by analyzing the long-term behavior of the function.

We wish to design a container for a limited edition of a premium perfume. The container must be shaped as a cylinder topped with a glass hemisphere. The total volume of the container must be exactly $250 \text{ cm}^3$. The cost of the material for the base and the cylindrical side is $2 \text{ \$/cm}^2$, while the glass for the top hemisphere, being decorated, costs $5 \text{ \$/cm}^2$.

Calculate the dimensions (radius $r$ and height of the cylinder $h$) that the container must have so that the manufacturing cost is a minimum. (Remember that the volume of a sphere is $V = \frac{4}{3}\pi r^3$ and its surface area is $S = 4\pi r^2$).

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

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

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

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 platform located on the x-axis, and from there continue towards $B$.

However, the energy consumption per kilometer flying over the platform is $25\%$ lower than flying directly towards it. The total energy cost function is $E(x) = \sqrt{x^2 + 16} + 0.75\sqrt{(10-x)^2 + 4}$. Find the exact position of the point $P$ that minimizes the total energy consumption.

A farmer has $120$ meters of fencing to enclose a piece of land divided into three equal, adjoining rectangular plots, as shown in the figure. The farmer wants to maximize the total enclosed area. Furthermore, the cost of the outer fence is $200 \text{ \$/m}$, but the fence used for the internal partitions is lighter and costs only $80 \text{ \$/m}$.

If the total budget for the fencing is $\$20\,000$, calculate the dimensions of each plot so that the total area is as large as possible. Note that here the constraint is not the total length of fencing, but the total budget available.

Consider the total manufacturing cost function of an electronic component $C(x) = \frac{1}{3}x^3 - 6x^2 + 40x + 100$, where $x$ represents the number of units produced in thousands. The sales revenue function is $I(x) = 28x - x^2$.

1. a) Determine the production interval in which the company makes a profit (where $I(x) > C(x)$).
2. b) Calculate the number of units that maximises monthly profit and the value of this maximum profit.
3. c) For what value of $x$ is the average cost per unit, defined as $Q(x) = \frac{C(x)}{x}$, at a minimum?

A logistics center is located at a strategic point whose daily truck traffic flow is modeled by the function $T(t) = t^3 - 15t^2 + 72t + 50$, where $t$ is the time of day in hours ($0 \leq t \leq 24$).

Calculate the times at which the maxima and minima (relative and absolute) of the truck flow occur during the working day. If the center's management decides to carry out maintenance work when traffic is decreasing most rapidly (minimum slope), at what time should it schedule these tasks? (Hint: look for the point of inflection of $T(t)$).

A window is in the shape of a rectangle surmounted by an equilateral triangle. The total perimeter of the window (including the base of the triangle separating the two parts) is $12 \text{ meters}$.

Calculate the dimensions of the rectangle (base $x$ and height $y$) so that the total area of the window is maximum and the light entering is as great as possible. (Recall that the area of an equilateral triangle with side $x$ is $A = \frac{\sqrt{3}}{4}x^2$).