Exercises: Function optimisation

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?

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.

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.

An artisanal footwear workshop in Elche has monthly production costs (in hundreds of euros) modeled by the function $C(x) = x^3 - 6x^2 + 15x + 10$, where $x$ represents the number of pairs of shoes made (in hundreds of units). Due to personnel and machinery constraints, the maximum monthly production is $400$ pairs, that is, the admissible physical domain is $x \in [0, 4]$.

Marcos, the trainee accountant, looks for the number of pairs that minimizes the total cost and writes in his report:

"I calculate the derivative $C'(x) = 3x^2 - 12x + 15$. When solving the equation $3x^2 - 12x + 15 = 0$, the discriminant is $(-12)^2 - 4 \cdot 3 \cdot 15 = 144 - 180 = -36 < 0$, so there are no real roots. Since the derivative does not vanish at any point, the function has no minimum and, therefore, the cost does not depend on how many shoes we manufacture."

1. a) Explain Marcos's error in reasoning in detail. What type of candidates has he omitted in his analysis and why is his conclusion regarding the independence of cost false?
2. b) Study the sign of $C'(x)$ on the entire interval $[0, 4]$ and deduce the monotonicity of the cost function with reasoning.
3. c) Determine the actual production $x \in [0, 4]$ that minimizes the monthly cost, indicating what said minimum cost is and what the cost would be at the opposite end of the domain.

A bus company operating the line between Granada and Motril currently charges €$15$ per ticket, recording an average of $180$ passengers per trip. The sales management considers offering discounts to attract more demand: for each €$0,50$ discount on the ticket ($x$ discounts of €$0,50$, so that the total discount is $0,5x$ euros), the number of passengers increases by $10$ people. The maximum capacity of the vehicle is $250$ passengers, so at most $70$ additional passengers can join ($0 \le x \le 7$).

1. a) Estimate: Before calculating derivatives, compute the current revenue (without discount, $x=0$) and the revenue if the maximum permitted discount were applied ($x=7$). From these two initial values, make a reasoned estimate of whether it would be advisable to apply the maximum discount, stay as is, or look for an intermediate price.
2. b) Calculate: Formulate the revenue function $I(x)$ in terms of the number of reductions $x$. Use the optimization procedure with derivatives to find the exact value of $x$ that maximizes revenue in the interval $[0, 7]$.
3. c) Check: Determine the optimal final ticket price, the resulting number of passengers, and the total revenue obtained. Compare these results with your initial estimate and assess whether the discount strategy is advisable compared to the previous situation.

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 $13,5$ litres ($13,5\ \text{dm}^3$). Determine the dimensions of the base side ($x$) and height ($h$) that minimise 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 $C(x) = 0,2x^2 + 10x + 500$ euros. The unit selling price depends on demand and follows the formula $p(x) = 70 - 0,1x$. 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 euros) following the function $S(x) = \frac{48}{x} + x - 10$ for prices between $4$ and $15$ euros. Calculate the subscription price that minimises 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 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 "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 euros follows the function $I(p) = -2p^2 + 120p$, where $p$ is the average price in euros of its garments. On the other hand, its daily expenses are modelled 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.

An olive oil cooperative in Jaén models its total packaging costs (in thousands of euros) using the function $C(q) = 0,1q^3 - 0,6q^2 + 4q + 10$, where $q$ is the volume packaged in thousands of liters, with $q \in [1, 8]$.

To find the production $q$ that minimizes the average cost per liter, $CM(q) = \dfrac{C(q)}{q}$, the technical team debates between two approaches:

• Method 1 (Direct optimization): Write the algebraic expression of $CM(q)$, calculate its derivative $CM'(q)$, set it equal to zero, and justify the minimum using the second derivative.

• Method 2 (Marginal-average property): Use the theoretical property according to which the average cost is minimized when it equals the marginal cost, solving the equation $C'(q) = CM(q)$ directly.

1. a) Develop both methods step by step until obtaining the equation that yields the candidate for the minimum. Prove algebraically, applying the quotient rule to $CM(q) = \dfrac{C(q)}{q}$, why both methods necessarily lead to the same equation.
2. b) Compare the computational effort of both methods: what algebraic advantage does Method 2 have over Method 1 when dealing with the linear term ($4q$) and the constant term ($10$) of the cost?
3. c) Solve the resulting equation, rule out any non-admissible roots, and verify that it is a local minimum using $CM''(q)$. State the optimal production $q$ and the resulting minimum average cost per liter (in euros).

An express parcel delivery company operates with electric delivery vans within a Low Emission Zone (ZBE) in Madrid. The operating cost per kilometer traveled, $C(v)$ in euros, depends on the average driving speed $v$ (in km/h) through three components:

• Battery electrical energy consumption: €$0,02v$/km.

• Driver's wage (€$18$/hour): since at an average speed $v$ it takes $\dfrac{1}{v}$ hours to travel one kilometer, this cost amounts to €$\dfrac{18}{v}$/km.

• Mechanical wear and vehicle depreciation: a fixed cost of €$0,10$/km.

1. a) Express the total cost function per kilometer $C(v)$. If initially traffic conditions allow driving at average speeds between $10$ and $60$ km/h ($v \in [10, 60]$), calculate the average speed that minimizes the cost per kilometer and justify that it is an absolute minimum on that interval.
2. b) Determine the minimum cost per kilometer incurred by the company driving at that optimal speed.
3. c) Due to a modification in the municipal urban pacification ordinance, the maximum speed in the district is restricted to $20$ km/h, so that the new operating domain becomes $[10, 20]$. Without the need to compute derivatives again, use the sign of $C'(v)$ on that interval to argue what the new adopted speed should be, and calculate the percentage of additional cost per kilometer that the company will suffer compared to the initial situation.

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 $\frac{C(t)}{100}$.

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 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.

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 farmer wants to fence off a piece of land divided into three equal, adjacent rectangular plots, as shown in the figure. The dimensions of each plot are $x$ (width) and $y$ (length), in meters. The farmer wants to maximize the total enclosed area.

The cost of the outer fence is $10\ \text{€/m}$, but the fence used for the two internal dividers is lighter and costs only $4\ \text{€/m}$. The total budget for the fence is $1000\ \text{€}$, and the only constraint is this total available cost.

Calculate the dimensions of each plot so that the total area is as large as possible.

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 centre is located at a strategic point whose daily truck traffic flow is modelled 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 centre's management decides to carry out maintenance work when traffic is decreasing most rapidly (minimum gradient), 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 $12 \text{ metres}$.

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 $A = \frac{\sqrt{3}}{4}x^2$).

A microelectronics company in Valencia needs to determine the parameters of the monthly total cost model for a new processor: $C(x) = a x^3 + b x^2 + c x + 54$ (in thousands of euros), where $x$ is the quantity produced in thousands of units ($x > 0$).

The finance department imposes two efficiency conditions:

1. The average cost per unit, $CM(x) = \dfrac{C(x)}{x}$, must reach its absolute minimum at a production level of $x = 3$ thousand units.

2. At said optimum ($x = 3$), the average cost must be exactly $35$ thousand euros for every thousand units produced ($CM(3) = 35$).

To find the three coefficients ($a, b, c$), an analyst suggests as a third condition: "Since at the minimum of average cost it coincides with marginal cost, we will also require that $C'(3) = 35$."

1. a) Prove analytically why the analyst's suggestion is mathematically useless because it is redundant: prove that any differentiable cost function that simultaneously satisfies $CM'(3) = 0$ and $CM(3) = 35$ will always and automatically satisfy $C'(3) = 35$.
2. b) To have a genuinely independent condition, the plant department provides a verified experimental data point: when production is $2000$ units ($x = 2$), the marginal cost is $20$ thousand euros for every thousand units ($C'(2) = 20$). Set up the corresponding system of equations and calculate the exact values of the parameters $a$, $b$, and $c$.
3. c) With the parameters obtained, verify using the second derivative that the average cost function truly has a minimum at $x = 3$, and check whether the total cost function $C(x)$ is strictly increasing for all $x > 0$.

A railway operator markets a high-speed charter train with a maximum capacity of $300$ passengers intended for end-of-year school trips between Madrid and Valencia. The fare policy establishes a base price of €$80$ per ticket for a minimum group of $100$ passengers. To encourage the booking of larger groups, for each student added above $100$ (that is, $x$ additional students, with $100 + x \le 300$), a discount of €$0,20$ is applied to the price of all tickets on the train. However, due to profitability policy, the maximum discount cannot exceed €$35$ per ticket (once that discount is reached at $x = 175$, the price is frozen at €$45$ for any larger group until filling the train).

The operating cost borne by the company consists of a fixed fee of €$3000$ per trip plus a variable cost of €$12$ for each passenger on board.

1. a) Formulate the net profit function $B(x)$ in euros, defining it as a piecewise function on the complete domain $x \in [0, 200]$, distinguishing the section with variable discount ($0 \le x \le 175$) from the section with frozen price ($175 < x \le 200$).
2. b) Analyze all candidates for the optimum: calculate the interior critical point in the first section using derivatives, justifying whether it is a local maximum; study the behavior of the profit in the second section through its derivative; and evaluate the profit at the boundary points ($x = 0$, $x = 175$, and $x = 200$).
3. c) How many tickets in total must the company sell to achieve the absolute maximum net profit, and what is its amount? Explain why an analyst who limits themselves to searching only for points where the derivative vanishes ($B'(x) = 0$) would make an erroneous and clearly suboptimal business decision.

Topic challenge. Design the optimal fee for a youth service managed by your local town council.

1. Phase 1: Sampling and inference. Ask $30$ people in your environment how much they would pay per month for a cultural/sports pass (between $5$ and $30$ €). With your data, calculate the sample mean and determine a $95\,\%$ confidence interval for the population mean price $\mu$.

2. Phase 2: Modelling. Assume a linear demand function $p(x) = \mu - 0,15x$, where $x$ is the number of users. Formulate the revenue function $I(x) = x \cdot p(x)$ and define a profit function $B(x) = I(x) - (40 + 2x)$ with estimated fixed and variable costs.

3. Phase 3: Optimisation with derivatives. Use GeoGebra or a spreadsheet to calculate the derivative function $B'(x)$ and the higher-order derivatives ($B''(x)$). Analyse the monotonicity of the function, locate the relative extrema, and justify whether the critical point obtained is a relative maximum or minimum.

4. Phase 4: Final product. Prepare a one-page report with the graph of $B(x)$, explaining which fee maximises profits and how the margin of error of the confidence interval affects it.

Assessment criteria:

• Precision in the calculation and statistical interpretation of the confidence interval.

• Rigorous application of higher-order derivatives to deduce monotonicity and relative extrema.

• Skill in mathematical representation and optimisation using GeoGebra or a calculator/spreadsheet.

• Clarity and socioeconomic coherence in the final proposal.