Exercises: Optimisation and complete study of functions

Let $f(x) = x^{2} - 4x + 3$.

1. a) Calculate $f'(x)$ and find the point where it equals zero.

1. b) Is this point a maximum or a minimum? Justify your answer by studying the sign of $f'$ on both sides.

The derivative of a function is $f'(x) = 2x - 6$.

1. a) At which point does $f'$ equal zero?

1. b) Where is $f$ increasing and where is it decreasing?

1. c) What type of extremum does $f$ have, and at which point?

The figure shows the weekly profit of an ice cream shop as a function of the ice cream price: the parabola intersects the price axis at €5 and at €25.

1. a) Using the symmetry of the parabola, which price maximises profit?

1. b) For which prices is the profit zero? What do these values mean?

1. c) For which prices is the profit negative? Interpret this.

The derivative of a function $f$ equals zero at $x = 2$ and at $x = 5$, and its sign is given by the table:

• In $(-\infty, 2)$: $f'(x) > 0$.

• In $(2, 5)$: $f'(x) < 0$.

• In $(5, +\infty)$: $f'(x) > 0$.

Classify the points $x = 2$ and $x = 5$ as relative maximum or relative minimum, justifying your answer.

Let $f(x) = x^{3} - 3x$.

1. a) Calculate $f'(x)$.

1. b) At which points does the derivative equal zero?

1. c) State the intervals where $f$ is increasing.

The profit of a factory, in thousands of euros, when producing $x$ tonnes per week is $B(x) = -2x^{2} + 80x - 300$.

1. a) What weekly production level maximises profit?

1. b) What is this maximum profit, in euros?

Determine whether the statement is true or false: "if $f'(3) = 0$, then $f$ has a relative maximum or minimum at $x = 3$". If false, give a specific counterexample and explain it.

There are $40$ m of fencing available to enclose a rectangular pen on all four sides.

1. a) If one side measures $x$, express the area of the pen in terms of $x$.

1. b) What dimensions maximise the area? What shape is the pen then?

An agricultural cooperative in Galway has $40\text{ metres}$ of wire fencing to enclose a rectangular vegetable garden that uses an already built straight stone wall as one of its sides (so it only needs to fence the other three sides). Due to requirements of the adjacent irrigation ditch, the side parallel to the wall cannot measure more than $15\text{ metres}$.

Seán, one of the partners, carries out the following calculations to maximize the area of the garden:

1. He calls $x$ the length of the two sides perpendicular to the wall and $y$ the parallel side: $2x + y = 40 \implies y = 40 - 2x$.

2. He expresses the area: $A(x) = x \times (40 - 2x) = 40x - 2x^2$.

3. He differentiates and sets it equal to zero: $A'(x) = 40 - 4x = 0 \implies x = 10\text{ m}$.

4. He concludes: "The perpendicular sides must measure $10\text{ m}$ and the parallel side $20\text{ m}$, yielding a maximum area of $200\text{ m}^2$".

1. a) Locate and explain with reasoning the conceptual error that Seán has made in the optimization process, indicating which fundamental step he did not take into account.
2. b) Establish the admissible domain of the variable $x$ taking into account the legal limitation imposed by the irrigation ditch and the real geometric conditions.
3. c) Correct the exercise: analyze the behavior of the function $A(x)$ on its real domain and determine the exact dimensions that maximize the area while respecting regulations, as well as the value of said maximum area.

A stall on Grafton Street in Dublin sells flower bouquets for Valentine's Day. The daily net profit $B(p)$, in euros, as a function of the selling price $p$ of each bouquet (in euros), is modeled by the quadratic function:

$B(p) = -20p^2 + 360p - 1,000$, for $p \in [5, 15]$.

1. a) Estimate: Without computing derivatives, find the two prices for which the net profit is zero ($B(p) = 0$). Knowing that the graph of any quadratic function is symmetric with respect to its vertex, deduce with reasoning at what price $p$ the maximum profit will be reached and estimate the value of said profit.
2. b) Check: Apply the method of differential calculus (first derivative, relative extremum condition $B'(p) = 0$, and second derivative test or sign of $B'$) to analytically calculate the optimal price and the maximum profit.
3. c) Compare both results, check if they match, and reflect: why in a quadratic polynomial function does the geometric estimation by symmetry yield exactly the same result as the derivative?

We want to fence $100$ m to make a rectangular vegetable patch against a wall, so that only three sides need to be fenced, as shown in the figure.

1. a) If $x$ is the length of each side perpendicular to the wall and $y$ is the length of the parallel side, express $y$ as a function of $x$ and write the area function $A(x)$.

1. b) Calculate the dimensions that maximise the area.

1. c) What is this maximum area?

From a square sheet of cardboard with side length $24$ cm, four identical squares of side length $x$ are cut out from the corners to construct, by folding, an open-topped box, as shown in the figure.

1. a) Express the volume of the box $V(x)$ as a function of $x$ and state its domain.

1. b) Calculate $V'(x)$ and find its critical points, discarding those that do not belong to the domain.

1. c) What value of $x$ maximises the volume? What is this maximum volume?

A company has a revenue $I(q) = -0.2q^{2} + 60q$ and costs $C(q) = 20q + 800$, in euros, where $q$ are the units produced and sold. The figure shows both curves.

1. a) Write the profit function $B(q)$ and calculate its derivative.

1. b) What production level maximises profit? What is this maximum profit?

1. c) Calculate the break-even points (the values of $q$ for which $B(q) = 0$) and relate them to the intersection points in the figure.

Study and sketch the function $f(x) = x^{3} - 3x^{2} + 2$:

1. a) Axes intercepts.

1. b) Intervals where the function is increasing and decreasing, and local extrema.

1. c) Curvature and point of inflection.

1. d) Sketch the graph.

You can do it on paper and send a photo with your answer.

Study the function $f(x) = \frac{x}{x^{2} + 1}$:

1. a) Domain and symmetry (check that it is odd).

1. b) Horizontal asymptote (calculate the limit as $x \to +\infty$).

1. c) Relative extrema and monotonicity.

1. d) Using the above information, sketch its complete graph.

You can do it on paper and send a photo with your answer.

A canning company wants to manufacture a box with a square base and a lid, with a volume of $1\ \text{dm}^{3}$ ($1,000\ \text{cm}^{3}$), using the minimum amount of cardboard.

1. a) If $x$ is the side length of the base and $y$ is the height, express $y$ in terms of $x$ using the volume condition.

1. b) Write the total surface area function $S(x)$ and calculate its derivative.

1. c) Find the dimensions that minimise the surface area. What shape is the optimal box?

The figure shows the graph of a polynomial function $f$. Read the values you need from the grid.

1. a) State the intervals where the function is increasing and decreasing.

1. b) What is the sign of $f'(x)$ in each of those intervals? What is the value of $f'$ at the points where the function changes from increasing to decreasing or from decreasing to increasing?

1. c) What are the coordinates of the relative maximum and minimum? Are they also absolute? Justify your answer by looking at the branches of the graph.

A market study estimates that the weekly profit of a shop, in euros, when selling $x$ units of a product is $B(x) = -x^{2} + 140x - 3,600$.

1. a) How many units must it sell to obtain maximum profit? How much is this profit?

1. b) For which sales levels does the shop lose money?

1. c) If, due to storage limitations, it cannot sell more than $50$ units, what is its best decision then? Justify your answer using the monotonicity of $B$.

An express parcel delivery company in Dublin analyzes the sorting rate of shipments in its logistics center during an uninterrupted shift of $8\text{ hours}$. The number of parcels sorted per hour is modeled by a cubic function:

$f(t) = a t^3 + b t^2 + c t$, with $t \in [0, 8]$ (where $t$ is the elapsed time in hours from the start of the shift).

The operations management has only three technical reports from the shift:

1. The sorting rate reaches a relative maximum $2\text{ hours}$ after starting the shift ($t = 2$).

2. The rate experiences its inflection point (change from acceleration to deceleration) exactly $4\text{ hours}$ into the shift ($t = 4$).

3. At the moment of peak activity ($t = 2$), exactly $32\text{ packages per hour}$ are being sorted.

1. a) Translate each of the three pieces of information into mathematical conditions on $f(t)$, $f'(t)$, or $f''(t)$ and set up a system of equations to determine the parameters $a$, $b$, and $c$.
2. b) Solve the system and write the complete analytical expression of $f(t)$.
3. c) From the function obtained, carry out the study of monotonicity over the entire interval $[0, 8]$ and determine at what times of the shift the sorting rate reaches its absolute minimum value, indicating how many parcels per hour are processed at those moments.

The municipal urban transport company of an Irish city such as Cork currently charges $1.20\text{ €}$ for a single bus ticket, with an average ridership of $10,000$ daily users. Mobility technicians estimate that, for every $0.10\text{ €}$ increase in the ticket price, $500$ daily users will be lost; and similarly, for every $0.10\text{ €}$ decrease, $500$ daily passengers will be gained. According to municipal regulations, the ticket price cannot be less than $0.60\text{ €}$ nor greater than $2.20\text{ €}$, and the maximum operational capacity of the fleet is $14,000$ daily passengers.

1. a) Let $p$ be the final price of the ticket in euros. Express the number of daily users $N(p)$ as a linear function of $p$, and deduce the physically admissible domain of $p$ taking into account the price restrictions and the maximum capacity of the fleet.
2. b) Write the daily revenue function $I(p)$ as a function of the price $p$. Justify why the revenue does not grow indefinitely as the price increases.
3. c) Use the tools of differential calculus to calculate the price $p$ that maximizes the daily revenue of the municipal company. Check that the obtained value belongs to the admissible domain, verify that it is indeed a maximum, and calculate the resulting maximum daily revenue.

Carry out a complete study of the function $f(x) = x^{3} - 12x$ and sketch its graph:

1. a) Domain, axis intercepts and symmetry.

1. b) Intervals where the function is increasing and decreasing, and local extrema.

1. c) Curvature and point of inflection.

1. d) Sketch the graph using the information obtained.

You can do it on paper and send a photo with your answer.

A cinema with $400$ seats sells tickets for $\text{€}\ 8$ and fills every session. The manager estimates that for every $\text{€}\ 0.50$ increase in price, they will lose $20$ viewers.

1. a) If $x$ is the number of $\text{€}\ 0.50$ price increases, write the revenue function $I(x)$.

1. b) What ticket price maximises the revenue? How much is this maximum revenue?

1. c) Above what price would the revenue be less than the current full house revenue?

1. d) The manager decides not to raise the price by more than $\text{€}\ 2$ so as not to empty the auditorium. Is it worth raising it by those $\text{€}\ 2$? Justify your answer using the model.

The total cost of producing $q$ units of a product is $C(q) = q^{2} + 30q + 25$, in euros.

1. a) Write the average cost per unit function, $CM(q) = \frac{C(q)}{q}$, for $q > 0$.

1. b) Calculate the value of $q$ that minimises the average cost and verify, by studying the sign of $CM'(q)$, that it is a minimum.

1. c) Calculate the marginal cost $C'(q)$ at that same value and check that it coincides with the minimum average cost. Is it a coincidence? Investigate the relationship between $CM'(q)$ and $C'(q) - CM(q)$ and state the general conclusion.

Find the point on the parabola $y = x^{2}$ closest to the point $P = \left(0, \frac{3}{2}\right)$.

1. a) Express the square of the distance from a generic point $(x, x^{2})$ on the parabola to the point $P$.

1. b) Differentiate this expression and find its critical points.

1. c) Determine which points on the parabola are at a minimum distance from $P$ and calculate that minimum distance. Why are there two solution points?

A window has the shape of a rectangle surmounted by a semicircle (arched window). The total perimeter of the frame is $12$ m and it is desired to let in as much light as possible, that is, the area should be maximum.

1. a) If $r$ is the radius of the semicircle and $h$ the height of the rectangle, write the relation between $r$ and $h$ imposed by the perimeter.

1. b) Express the total area $A(r)$ as a function of $r$ only.

1. c) Find the value of $r$ that maximises the area, leaving the result in terms of $\pi$, and calculate the full dimensions of the window.

A student solves an optimisation problem as follows: "We have to divide $50$ m of string into two pieces to enclose two square areas of side $x$ and side $y$. The total area is $A = x^{2} + y^{2}$. Since $4x + 4y = 50$, it follows that $y = 12.5 - x$, so $A(x) = x^{2} + (12.5 - x)^{2}$. Differentiating: $A'(x) = 4x - 25 = 0$, hence $x = 6.25$. Therefore, the maximum area is $6.25\ \text{m}^{2}$."

1. a) Find the two errors in their conclusion (one regarding the value of $6.25$ and another regarding whether it is a maximum or a minimum).

1. b) Solve the problem correctly: how should the string be cut to minimise the total area? And to maximise it? (Think about what happens at the boundaries of the domain of $x$.)

Reasonably assess whether these statements are true or false, providing examples to justify your answer:

1. a) "Every continuous function on a closed interval $[a, b]$ achieves a maximum value and a minimum value on it."

1. b) "Every continuous function on an open interval $(a, b)$ achieves a maximum value and a minimum value on it."

1. c) "If a function has a local maximum at $x = c$, that is also its absolute maximum over its entire domain."

Find the absolute maximum and minimum values of the function $f(x) = x^{3} - 6x^{2} + 9x + 1$ on the interval $[0, 4]$.

1. a) Calculate the critical points inside the interval.

1. b) Evaluate the function at the critical points and at the endpoints of the interval.

1. c) Conclude: indicate the absolute maximum and minimum and at which points they are reached. Are you surprised by any of the results? Comment on them.

The figure shows the graph of a rational function with a vertical asymptote $x = 2$ and a horizontal asymptote $y = 1$.

1. a) State the domain of the function and calculate, from the graph, $\lim_{x \to 2^{-}} f(x)$, $\lim_{x \to 2^{+}} f(x)$ and $\lim_{x \to +\infty} f(x)$.

1. b) Estimate the coordinates of the local maximum marked with "?" and explain what condition the derivative must satisfy at that point.

1. c) State the intervals where each branch is increasing and decreasing, and reason why the function has no extrema on the right branch.

1. d) A classmate claims that the function could be $g(x) = 1 + \frac{3}{x - 2}$, which has the same asymptotes. Calculate $g'(x)$, study its sign and reason why that function cannot be the one in the graph.

An olive-growing cooperative in Cork produces extra virgin olive oil. The total cost (in hundreds of euros) of bottling $x$ hundreds of liters of oil is given by the function:

$C(x) = x^3 - 2x^2 + 10x + 8$, for $x \in [1, 5]$.

The average unit cost per hundred liters produced is defined as $\overline{C}(x) = \frac{C(x)}{x} = x^2 - 2x + 10 + \frac{8}{x}$.

The financial director and the head of production discuss how to find the production volume $x$ that minimizes the average cost:

• Method 1 (Direct path): Differentiate the average cost function $\overline{C}(x)$ directly, set it equal to zero $\overline{C}'(x) = 0$, and solve the resulting equation.

• Method 2 (Principle of marginal economics): Equate the marginal cost to the average cost, that is, solve the equation $C'(x) = \overline{C}(x)$, based on the fact that the average cost stabilizes when the cost of producing an additional unit coincides with the current average.

1. a) Formally prove that for any differentiable cost function $C(x)$, the condition $\overline{C}'(x) = 0$ is mathematically equivalent to the equality $C'(x) = \overline{C}(x)$.
2. b) Carry out both methods for the cooperative's specific function. Compare the algebraic steps of both paths and justify which of the two methods is more efficient or less error-prone when differentiating quotients.
3. c) Solve the equation in the interval $[1, 5]$ to find the optimal production level $x$ and the corresponding minimum average cost. Justify by analyzing the sign of the derivative at the endpoints of the interval that the value obtained is an absolute minimum.

A rural tourism operator in County Kerry analyzes the weekly net revenue $I(t)$ (in thousands of euros) generated by its booking website for country cottages, as a function of the lead time $t$ (in weeks) with which tourists book their stay for the summer season ($0 \le t \le 10$). The estimated model corresponds to the function:

$I(t) = \frac{50t}{t^2 + 16}$.

1. a) Carry out a complete analytical study of the function $I(t)$ on its domain $[0, 10]$:

1. Axis intercepts and sign of the function.

2. Intervals of increase, decrease, and relative extrema (by computing $I'(t)$).

3. Concavity, convexity, and inflection points (by computing $I''(t)$).

4. Behavior at the endpoints of the interval and determination of the horizontal asymptote if the lead time could grow indefinitely ($t \to +\infty$).

1. b) Determine the optimal lead time with which the operator achieves maximum weekly revenue, calculating said amount in euros.
2. c) Economic interpretation: Explain the practical meaning of the inflection point found in section a) in relation to the rate at which revenue decreases when bookings are made too far in advance.

Topic challenge. At what time should a bakery turn on its ovens to minimise costs? Take on the role of an economic consultant and optimise the energy consumption of a local business.

1. Data collection: Visit the EirGrid website (or the SEMOpx market data) and collect the electricity price (€/MWh) for the 24 hours of the previous day. Enter them into GeoGebra or a spreadsheet and obtain a polynomial fit $C(t)$ for $t \in [0, 24]$.

2. Rate of change and derivative: Calculate the Average Rate of Change of $C(t)$ during off-peak and peak periods. Apply differentiation rules to find $C'(t)$ and interpret the marginal cost at two key hours.

3. Complete study and optimisation: Solve $C'(t) = 0$ and use $C''(t)$ to classify relative extrema, intervals of increase/decrease, and concavity. Determine the exact hour at which the cost is minimal.

4. Reasoned proposal: Write a brief report (one page) including the graph, the analytical development, and a justified schedule recommendation for the business.

Assessment criteria:

• Precision in calculating the average rate of change, differentiation rules, and extrema optimisation.

• Rigour in the complete analytical study and in the economic interpretation of marginal analysis.

• Quality of the final proposal and appropriate use of GeoGebra or the spreadsheet.