Exercises: Monotonicity of a function

Observe the following graph, which represents the evolution of the number of users of a new mobile app in its first 8 hours after launch. Identify, simply by observing the diagram, the time intervals in which the number of users is increasing and those in which it is decreasing.

Given the polynomial function $f(x) = x^2 - 6x + 8$, follow these steps to study its monotonicity:

1. Calculate its derivative function $f'(x)$.

2. Find the value of $x$ for which the derivative is equal to zero.

3. Determine the sign of the derivative in the intervals created by that point and state in which of them the function increases and in which it decreases.

A market study indicates that the daily profit of a small textile company (in hundreds of euros) is given by the function $B(t) = -t^2 + 10t - 16$, where $t$ is the time in hours since opening. Use the derivative to find in which time interval the company's profit is increasing.

Given the function $f(x) = 2x^3 - 6x$, we want to know how it behaves near the origin. Calculate the gradient of the tangent line (the derivative) at the points $x = -2$, $x = 0$ and $x = 2$. Based solely on the sign of these results, state whether the function is increasing or decreasing at each of these points.

Consider the function $f(x) = \frac{1}{3}x^3 - 4x + 5$.

1. Calculate $f'(x)$.

2. Solve the equation $f'(x) = 0$ to find the critical points.

3. Plot the points on the number line and study the sign of the derivative to write down the intervals of increase and decrease.

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

We know that a function $g(x)$ has derivative $g'(x) = x - 4$. Without needing to know the expression of $g(x)$, determine for which values of $x$ the original function $g(x)$ is increasing and for which it is decreasing. Justify your answer using the concept of derivative.

Check whether the linear function $f(x) = 5x - 3$ is always increasing across its entire domain. To do so, calculate its derivative and explain how the result obtained relates to the gradient of the line.

The height of a basketball (in metres) when thrown is modelled by the function $h(t) = -5t^2 + 6t + 2$, where $t$ is the time in seconds.

1. Calculate the instantaneous velocity of the ball by differentiating the height function.

2. Determine the time interval in which the ball is rising (increase of the height function).

In an olive oil cooperative in Jaén, the monthly net profit $B(x)$, expressed in thousands of euros, depends on the quantity $x$ of thousands of liters of extra virgin olive oil produced and bottled ($x \in [0, 8]$), according to the model:

Marcos, one of the partners, states the following at an assembly:

“I have calculated that the profit is zero for $x = 3$ and for $x = 7$. Since between $3$ and $7$ the profit is positive (for example, for $x = 5$ we make $4$ thousand euros), I claim that as long as we produce between $3$ and $7$ thousand liters, the company is in a phase of growing profits.”

a) [SOMEONE ELSE'S ERROR] Point out precisely the reasoning error that Marcos makes by confusing the sign of the profit with the sign of its first derivative (monotonicity).

b) Correct his analysis by calculating $B'(x)$ and deducing the actual interval on which the profit is strictly increasing and that on which it decreases.

The State Meteorological Agency (AEMET) records the temperature $T(t)$, in degrees Celsius, at a station in the province of Teruel during the daylight hours of a clear winter day ($t$ represents solar time, with $t \in [8, 18]$). The record fits the function:

a) [ESTIMATE AND CHECK] Before doing any mathematical calculation, make a reasoned estimate based on everyday experience: do you expect the temperature to be increasing or decreasing at $10:00$ in the morning ($t = 10$)? And at $17:00$ in the afternoon ($t = 17$)? Briefly justify your physical intuition.

b) Analytically check your prediction: calculate the derivative function $T'(t)$, evaluate its sign at $t = 10$ and at $t = 17$, and interpret the economic-environmental meaning of the signs obtained, indicating whether they confirm your initial estimate.

A streaming service company has found that the number of new daily subscribers, in thousands, follows the function $S(t) = -t^3 + 12t^2 + 45t + 100$, where $t$ represents the time in months since the start of an advertising campaign ($0 \leq t \leq 15$).

a) Determine the time intervals in which the growth of new subscribers is positive (the function is increasing).

b) From which month onwards does the trend change and the number of new daily subscribers starts to decrease? Justify the answer by studying the first derivative.

Study the monotonicity of the rational function $f(x) = \dfrac{x^2 + 9}{x}$. This type of function is often used to model average production cost in Social Sciences. State, with justification, the intervals where the average cost is decreasing and those where it is increasing, bearing in mind that in this context only values $x > 0$ make sense.

The figure shows the graph of $f$ for $x > 0$; you can use it to check your result.

Given the function $f(x) = x^3 + kx^2 + 15x + 2$, calculate the values of the parameter $k$ such that the function is strictly increasing across the entire domain of real numbers.

A sociological study analyses teenager interest in a new trend on social networks. The percentage of the population following the trend is given by the function $P(t) = \dfrac{50t}{t^2 + 4}$, where $t$ is the time in weeks ($t \geq 0$).

Determine in which time periods the trend is booming (increasing) and when it starts to lose interest among young people (decreasing).

The graph of the derivative function $f'(x)$ of a polynomial function $f(x)$ is shown below. Based on observing the graph of the derivative, state the intervals of increase and decrease of the original function $f(x)$.

Consider the piecewise-defined function:

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

Study the monotonicity of the function across its entire domain. Is there any point where the function changes from increasing to decreasing or vice versa? Justify whether there is a change in the behaviour of the function at $x=0$.

The profit of a tech startup, in hundreds of thousands of euros, is modelled by the function $B(x) = (x-2) \cdot e^{0,5x}$, where $x$ is the number of years since its founding ($x \geq 0$).

study the intervals of increase and decrease of the profit. For how long did the company record decreasing profits (or increasing losses) before starting to improve its financial situation on a sustained basis? If there was no such phase, justify why.

Given the function $f(x) = \dfrac{x-1}{x^2 + 3}$, determine its intervals of increase and decrease. From this study, state whether the function has any relative maximums or minimums, specifying the value of $x$ where they occur.

An express courier company that delivers in the metropolitan area of Seville estimates that the average operating cost per parcel, $C(x)$ (in euros), when distributing $x$ hundred parcels daily ($x \in [1, 8]$), corresponds to the formula:

To determine the activity range in which the cost per parcel decreases and above what volume it begins to become more expensive due to logistical saturation, two analysts debate which procedure to follow:

• Clara proposes evaluating the average cost at several integer values ($x = 2$, $x = 4$, $x = 6$) and deducing the trend from those specific numerical results.

• Roberto proposes calculating the derivative $C'(x)$, finding its zeros in the interval $[1, 8]$, and deducing the monotonicity by analyzing the sign of the derivative in each section.

a) [CHOOSE THE METHOD] Critically compare both methods. Explain why Clara's method can be misleading or insufficient to guarantee the overall behavior of the function, and justify why Roberto's method is mathematically sound.

b) Apply Roberto's method: find $C'(x)$, factor its numerator, study the sign of the derivative in $[1, 8]$, and determine with justification the intervals of decrease and increase of the average cost.

The Institute of Statistics of Andalusia publishes data on hotel occupancy in Costa del Sol. The number of monthly overnight stays, $P(t)$ (in thousands), over the course of a calendar year ($t \in [1, 12]$, where $t = 1$ represents January and $t = 12$ December), is approximated by the function:

a) [MODEL] Study the monotonicity of $P(t)$: calculate the first derivative, determine its zeros within the year, and identify the exact months in which hotel tourism is in a growth phase and those in which it is decreasing.

b) Calculate the numerical values of $P'(5)$ and $P'(10)$. Interpret both results in the real-world tourism context, explaining the difference between the rate of change in May ($t = 5$) and in October ($t = 10$), as well as the strategic situation of the sector in the month in which $P'(t) = 0$.

A production company is studying the audience evolution for a newly released documentary. The weekly number of viewers, in thousands, follows the function $R(t) = (t^2 + at + 10) \cdot e^{-0,5t}$, where $t$ is the time in weeks since the premiere ($t \ge 0$). Determine the value of the parameter $a \in \mathbb{R}$ so that the audience size is strictly decreasing during the first twelve weeks (interval $[0, 12]$) and describe how the sign of this parameter affects the rate at which the audience drops.

The following graph represents the derivative function $f'(x)$ of a continuous function $f(x)$ defined on all of $\mathbb{R}$. Based on observing the graph of the derivative, carry out a comprehensive study of the monotonicity of the original function $f(x)$, indicating its intervals of increase, decrease, and the x-coordinates where its relative maximums and minimums are reached. Justify whether it is possible for the function $f(x)$ to be a third-degree polynomial based on the number of changes in monotonicity.

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

A courier company has determined that the daily operating cost of its fleet of electric vehicles depends on the average speed $v$ (in km/h) according to the function $C(v) = \frac{2v^2 + 450}{v-10}$ for speeds above $10 \text{ km/h}$.

1. a) Study the monotonicity of the cost function and state the speed above which the cost starts to become inefficient (when the cost starts increasing again).
2. b) If traffic regulations require the average speed to lie within the interval $[20, 60]$, determine the intervals of increase and decrease within that restricted domain.

Consider the function depending on two parameters $a$ and $b$:

$f(x) = \begin{cases} x^2 + ax + b & \text{if } x < 1 \\ \ln(x) & \text{if } x \ge 1 \end{cases}$

Find the values of $a$ and $b$ such that the function is differentiable at the point $x = 1$. Once obtained, study the monotonicity of the resulting function across its entire domain $\mathbb{R}$. Does the function exhibit any change in monotonicity at the joining point of the two branches? Justify your answer using one-sided derivatives.

The number of active users on a youth social network during the first 24 hours after the launch of a viral challenge is given by the function $U(t) = \frac{100t^2}{t^3 + 16}$, where $t$ is the time in hours.

1. a) Determine the time intervals in which the challenge is gaining popularity and those in which it is losing interest.
2. b) Find the exact moment when user growth is at its maximum (that is, when the growth rate $U'(t)$ reaches its highest value). Hint: Study the monotonicity of the derivative function.

Given the function $f(x) = \frac{e^x}{x^2 - k}$, study its monotonicity in terms of the values of the real parameter $k$. In particular, prove that if $-1 < k < 0$, the function has a single interval of decrease. For the case $k = 4$, determine all the intervals of increase and decrease, taking into account the discontinuities of the function.

A sociological study analyses the level of "collective stress" $S$ in a city during an 8-hour event. The function modelling it is $S(t) = |t^2 - 6t + 5| + 2t$, with $t \in [0, 8]$.

1. a) Express the function $S(t)$ as a piecewise-defined function.
2. b) Study the intervals of increase and decrease of the stress level.
3. c) Identify the points where the function is not differentiable and explain whether a change in monotonicity occurs at any of those points.

Consider the function $f(x) = x^2 - \ln(x^2 - 1)$.

1. a) Determine the domain of definition of the function.
2. b) Study the monotonicity of the function, clearly indicating the intervals of increase and decrease.
3. c) Prove with reasoned arguments that the function is always increasing for any value of $x$ greater than $\sqrt{2}$. What happens in the interval $(1, \sqrt{2})$? Explain the behaviour of the function near its vertical asymptotes in relation to its monotonicity.

A financial consulting firm designs a net profit model $B(x) = ax^3 + bx^2 + cx$ (in thousands of euros) for a Valencian technology components company, as a function of the production volume $x$ (in thousands of units, with $x \ge 0$). Management requires the model to satisfy three simultaneous conditions:

1. The marginal profit is zero exactly when $2$ thousand units are manufactured and when $6$ thousand units are manufactured.

2. The profit function is strictly decreasing on the interval $(2, 6)$ due to extraordinary storage costs, and strictly increasing on $[0, 2)$ and on $(6, +\infty)$.

3. For a production level of $x = 1$ thousand units, the marginal profit is $15$ thousand euros for each additional thousand units manufactured ($B'(1) = 15$).

a) [INVERSE PROBLEM] Reasonably translate each of the conditions into the language of derivatives and set up the system needed to find the values of the parameters $a$, $b$, and $c$. Prove that the resulting function strictly satisfies the required intervals of increase and decrease.

b) The operations director states: “Since the company is increasing again for any production greater than $6$ thousand units, at $x = 7$ we will certainly make more total profit than at any point in the falling section $(2, 6)$.” Refute or confirm his statement by calculating $B(3)$ and $B(7)$, and argue why monotonicity describes the direction of local change and not the absolute value of profit.

The Ministry for the Ecological Transition and the Demographic Challenge studies the feasibility of a pilot plan against depopulation in a rural district. The socioeconomic dynamism index $I(t)$ over time ($t \ge 0$, measured in years from the start of the plan) responds to the model:

where $k \ge 0$ is a parameter that measures the budgetary intensity of public aid allocated to the district.

a) [MODEL] Formally prove that if no funds are allocated ($k = 0$), the derivative function $I'(t)$ is negative for all $t > 0$. Conclude and justify what will happen to the district's dynamism in the absence of investment.

b) By calculating the general derivative $I'(t)$, analyze the value of $I'(0)$ for any investment level $k > 0$. Relying on the continuity of derivative functions, explain why it is guaranteed that, for any $k > 0$, there will always be an initial period of revitalization (growth).

c) If a financial endowment that sets the parameter to $k = 3$ is approved, solve the equation $I'(t) = 0$, determine the intervals of increase and decrease of the index on $[0, +\infty)$, and explain in which exact year it will achieve its highest positive impact before starting to lose dynamism.