Exercises: Relative extrema

Observe the graph of the function $f(x)$ representing the evolution of the number of active users on a new mobile app during the first 8 hours of the morning. Identify the coordinates $(x, y)$ of the relative maxima and minima shown in the diagram.

Given the profit function of a small textile company, defined by $B(x) = -x^2 + 10x - 16$, where $x$ is the number of garments manufactured (in hundreds):

1. Calculate the derivative function $B'(x)$.

2. Find the value of $x$ that makes the derivative zero (critical point).

3. Determine whether this point corresponds to a relative maximum or minimum.

An influencer has found that the number of "likes" on their posts follows the function $L(t) = t^3 - 3t^2 + 4$, where $t$ is the time in hours since publication ($0 \le t \le 3$).

Calculate the critical points of the function in that interval by solving the equation $L'(t) = 0$.

Classify the relative extrema of the function $f(x) = x^2 - 6x + 8$ using the second derivative test. To do this:

1. Find $f'(x)$ and solve for $x$ such that $f'(x) = 0$.

2. Find $f''(x)$ and check its sign at the value obtained above.

Given the following graph of the derivative function $f'(x)$ (a straight line), state for which values of $x$ the original function $f(x)$ has a relative maximum or minimum. Justify your answer based on the sign change of the derivative.

Find the relative extrema of the polynomial function $f(x) = \frac{x^3}{3} - 4x + 5$. State the full coordinates $(x, y)$ of each identified extremum.

It is known that a function $f(x)$ has a critical point at $x = 3$. If the second derivative is $f''(x) = 2x - 10$, use the value of $f''(3)$ to determine whether there is a relative maximum or a relative minimum at $x = 3$.

The function $C(x) = 2x^2 - 8x + 20$ describes the maintenance costs of a web server as a function of the workload $x$.

Sketch the parabola and find the value of $x$ that minimises the cost.

SOMEONE ELSE'S MISTAKE. Mateo is analyzing the profitability of a small family business and states the following in his report:

"For the unit cost function $C(x) = x^3 - 6x^2 + 12x + 10$ (in euros, where $x$ is the number of batches produced), I have calculated $C'(x) = 3x^2 - 12x + 12$. Solving $C'(x) = 0$, I obtain a unique solution at $x = 2$. Since the derivative vanishes, at $x = 2$ there must necessarily be a relative extremum. Furthermore, calculating the second derivative I obtain $C''(2) = 0$, which confirms that it is not a maximum, so it has to be a relative minimum."

1. a) Identify the two conceptual errors that Mateo has made in his theoretical reasoning.
2. b) Correct the analysis: study the sign of $C'(x)$ to the left and to the right of $x = 2$ and determine in a reasoned manner what actually happens at that production point.

CHOOSE THE METHOD. An e-commerce company estimates that daily net revenue derived from an advertising campaign is given by the function $I(x) = -x^4 + 4x^3$ (in hundreds of euros), where $x$ represents the daily advertising investment (in hundreds of euros, with $x \ge 0$).

1. a) Find the two critical points of the function $I(x)$ by setting its first derivative equal to zero.
2. b) Try to classify both points using the second derivative test. What happens at each of them? Does this test allow you to classify both points, or is it inconclusive for either of them?
3. c) Apply the first derivative test (sign change) to study the behavior around the critical points. Based on the results, compare both methods: explain which one was more effective in this specific case and conclude what type of point (relative maximum, relative minimum, or neither) each one represents.

A content creator has analysed the evolution of views on their videos during the first 10 hours after publication. The function describing the cumulative number of views (in thousands) is given by $V(t) = -t^3 + 12t^2 + 60t$, where $t$ is the time in hours, with $t \in [0, 10]$.

Calculate the moment when the number of views is growing fastest. To do this, find the value of $t$ that maximises the growth rate function $V'(t)$ and determine the relative extrema of the original function $V(t)$ in that interval.

Calculate the values of the parameters $a$ and $b$ so that the polynomial function $f(x) = ax^3 + bx^2 + 9x + 4$ has a relative maximum at the point $P(1, 8)$. Once the values have been found, check using the second derivative whether it is indeed a maximum or a minimum.

The following graph corresponds to the derivative function $f'(x)$ of a continuous function $f(x)$. From the visual information, identify with reasons the x-coordinates (abscissae) of the points where the original function $f(x)$ has its relative maxima and minima. Explain what test or criterion you use to differentiate between a maximum and a minimum based on the sign of $f'(x)$.

A courier company has determined that the average cost of delivering a parcel depends on the distance $x$ (in km) to the distribution centre according to the function:

$C(x) = \dfrac{2x^2 + 32}{x}$ for $x > 0$.

Find the distance $x$ at which the delivery cost is a minimum. Justify that the value obtained is indeed a relative minimum using the first or second derivative test.

The effectiveness of a new drug for the study of memory in teenagers is measured on a scale from 0 to 20. After ingestion, the effectiveness $E$ after $t$ hours is given by the exponential function:

$E(t) = 10t \cdot e^{-0,2t}$ for $t \ge 0$.

Determine the time $t$ at which the drug reaches its maximum effectiveness. State the value of this maximum effectiveness and confirm whether it is a relative extremum.

Let the rational function be $f(x) = \dfrac{x^2 - 1}{x^2 + 1}$.

Study the increase and decrease of the function and locate its relative extrema. Does this function have any point where the derivative is zero but is neither a maximum nor a minimum? Give reasons for your answer.

Consider the piecewise function:

$f(x) = \begin{cases} x^2 + 4x + 3 & \text{if } x \le 0 \\ \dfrac{3}{x+1} & \text{if } x > 0 \end{cases}$

Study the continuity and differentiability at $x = 0$. Based on the previous study and on calculating the derivative in each section, determine whether the function has a relative maximum or minimum at $x = 0$.

A marketing department estimates that the monthly profits of an advertising campaign (in euros) follow the function $B(x) = -x^3 + 15x^2 + 600x - 1000$, where $x$ is the number of media channels used ($x \in [1, 30]$).

Find the number of channels that maximises the monthly profit. Distinguish between the relative extrema found inside the interval and the values that the function takes at the boundaries of the domain to ensure the absolute maximum.

ESTIMATE AND CHECK. In the regulated electricity market in Spain (PVPC tariff), the large contribution of photovoltaic solar generation usually reduces the hourly price of electricity in the central hours of the afternoon (usually between 14:00 and 18:00), before the price rises again at dusk. An analyst models the price of a megawatt-hour (in euros) on a spring day using the function $P(t) = 0,1t^3 - 4,2t^2 + 57,6t - 150$, where $t$ represents the hour of the day in the interval $t \in [10, 20]$ (from 10:00 to 20:00).

1. a) Before performing any derivative calculations, reasonably estimate in which time slot the relative minimum of prices should be located and justify whether you expect the price at 12:00 to be higher or lower than in the mid-afternoon.
2. b) Calculate exactly the critical points of $P(t)$ in the open interval $(10, 20)$ and classify them with justification using the second derivative test.
3. c) Check whether the relative minimum obtained matches your previous estimate and explain the usefulness of knowing the exact hours of the relative extrema for a company that needs to schedule intensive machinery consumption.

INVERSE PROBLEM. An urgent parcel delivery company in the metropolitan area of Madrid needs to design a model to predict its daily operating costs $C(x) = a x^3 + b x^2 + c x + 500$ (in euros), where $x$ is the number of delivery vehicles in circulation ($x \in [1, 15]$). The management team imposes the following three efficiency conditions:

1) The function must reach a relative maximum of inefficiency when exactly $x = 2$ vehicles are circulating (due to initial scale problems).

2) The function must reach a relative minimum of operating cost when exactly $x = 10$ vehicles are circulating (optimal route dispatch point).

3) The initial marginal cost for $x = 0$ (that is, the slope $C'(0)$) must be exactly $60$ euros per vehicle.

1. a) Set up the system of linear equations that allows calculating the parameters $a$, $b$, and $c$, and solve for their exact values.
2. b) With the calculated values, formally justify using the sign of $C'(x)$ or using the second derivative that at $x = 2$ there is a relative maximum and at $x = 10$ there is a relative minimum.
3. c) If the company must choose between keeping 2 vans or 10 vans operational, argue economically which is the correct decision based on the theory of relative extrema.

A digital marketing agency has determined that the engagement level of a social media campaign follows the function $E(t) = \frac{at}{1 + bt^2}$, where $t$ is the time in days since the launch. It is known that the maximum engagement level is reached exactly at 4 days and that, at that moment, the engagement value is 10 units.

1. a) Determine the values of the parameters $a$ and $b$.
2. b) Justify, using the second derivative test, that at $t = 4$ there is indeed a relative maximum.
3. c) What happens to the engagement in the very long term? Interpret the result.

Consider the profit function of a tech startup defined piecewise for the first two years ($t \in [0, 24]$ months):

$B(t) = \begin{cases} t^2 - 12t & \text{if } 0 \le t < 12 \\ \ln(t - 11) + k & \text{if } 12 \le t \le 24 \end{cases}$

1. a) Calculate the value of $k$ so that the function is continuous at $t = 12$.
2. b) For the value of $k$ found, study whether the function is differentiable at $t = 12$.
3. c) Determine the intervals of increase and decrease, and find the relative extrema (local maxima and minima) of the function over the entire interval.

A logistics study for a food delivery company estimates that the hourly operating cost depends on the average speed $v$ (in km/h) of its delivery drivers according to the function:

$C(v) = 2v^2 + \frac{16000}{v} + 50$ for $v > 0$.

1. a) Calculate the optimal speed that minimises the operating cost.
2. b) Show that there are no relative maxima for this function in its domain.
3. c) If, due to safety regulations, the average speed cannot exceed 15 km/h, what would be the actual minimum cost the company could achieve?

Given the polynomial function $f(x) = x^3 + ax^2 + bx + c$, find the values of $a$, $b$ and $c$ knowing that the function has a point of inflection at $x = 1$, a relative extremum at $x = 2$, and that the graph passes through the point $(0, 5)$. Once the parameters have been found, classify all the relative extrema of the function.

An "influencer" analyzes the virality of his videos. The number of daily views $V$ (in thousands) after $t$ hours since posting, with $0 \le t \le 24$, follows the model $V(t) = t^2 \cdot e^{-t/4}$. The figure shows the graph of the function.

1. a) At what exact moment of the day does the video reach its peak number of views?

1. b) Determine the time intervals in which the growth rate of the views is increasing (use the second derivative).

1. c) Calculate the maximum value of daily views.

Find the relative extrema of the function $f(x) = \frac{\ln(x^2)}{x}$. Pay special attention to the domain of the function and its symmetry to simplify the study. Justify whether the extrema found are absolute maxima or minima in the interval $(0, +\infty)$.

We wish to design a rectangular billboard with a surface area of $18$ $m^2$. The cost of the top and bottom frame is $20$ € per linear metre, while the cost of the sides is $15$ € per linear metre. Furthermore, for aesthetic reasons, the height of the billboard cannot be less than $2$ metres or greater than $5$ metres.

1. a) Express the total frame cost function in terms of the width $x$.
2. b) Determine the dimensions of the billboard that minimise the total cost within the given constraints.
3. c) What is this minimum cost?

Let the function be $f(x) = |x^2 - 4| \cdot e^x$.

1. a) Express the function without modulus signs.
2. b) Study the differentiability at the points $x = -2$ and $x = 2$.
3. c) Determine all relative maxima and minima of the function. State which of them correspond to points where the function is not differentiable (sharp points).

MODEL. The hotel occupancy rate $H(t)$ (in percentage, out of $100$) in a tourist region of the Pyrenees of Huesca over the 16 weeks spanning from the start of the ski season to the beginning of the spring routes ($t \in [0, 16]$, where $t$ is measured in weeks) follows a dual behavior modeled by the function:

1. a) Check whether the function $H(t)$ is continuous at $t = 8$ and analyze whether it is differentiable at that point by calculating the one-sided derivatives $H'(8^-)$ and $H'(8^+)$.
2. b) Determine all candidates for relative extrema in the open interval $(0, 16)$, clearly distinguishing between the points where the derivative vanishes and those points where the derivative does not exist.
3. c) Study the sign of the first derivative in each of the generated intervals and rigorously classify all the critical points found (indicating whether they are relative maxima, relative minima, or neither).
4. d) Interpret the results in the tourism context: in which week is the peak of the winter season recorded, and in which week is the bottom hit in the intermediate trough? Does the season changeover week ($t = 8$) constitute a relative extremum of occupancy? Justify your answer.

INVERSE PROBLEM WITH PARAMETER DISCUSSION. The financial department of a telecommunications company evaluates the projected net profit $B(x)$ (in thousands of euros) derived from launching a digital service depending on the investment in servers $x$ (in tens of thousands of euros, with $x \ge 0$). The mathematical model is given by the expression:

where $k$ is a positive real parameter reflecting the degree of infrastructure saturation.

1. a) Calculate the first derivative $B'(x)$ and discuss, according to the different values of $k > 0$, how many critical points the function has and for which values of $k$ relative extrema exist in the domain.
2. b) For the critical value $k = 3$, check what happens with the second derivative test at the resulting critical point and prove, by studying the sign of the first derivative, why neither a relative maximum nor a relative minimum exists at that point.
3. c) If market studies confirm that the company reaches a relative maximum of profits when the investment is €$10\,000$ ($x = 1$):

1. Determine the value of the parameter $k$.

2. Find the coordinates $(x, B(x))$ of all relative extrema (maximum and minimum) and classify them.

3. Reason about the behavior of profit: explain why after reaching the relative maximum at $x = 1$, moderately increasing the investment causes a drop in profits down to a relative minimum before growing again on a large scale.