Exercises: Intervals of increase and decrease

The following graph represents the change in the number of viewers (in thousands) of a newly released series on a streaming platform during its first 8 weeks. Visually identify the time intervals in which the audience has been growing and those in which it has been decreasing.

Based on the relationship between the derivative and growth, what sign do you think the derivative of this function would have on the interval from the start to point $A$ (week 3)?

A small phone case design company has determined that its daily profits, in euros, are given by the function $B(x) = -x^2 + 12x - 20$, where $x$ is the number of phone cases sold per day.

Calculate the derivative function $B'(x)$ and study for which values of $x$ the function is increasing and for which it is decreasing. After how many phone cases sold do profits start to decrease?

Given the polynomial function $f(x) = x^3 - 3x + 5$, carry out the following steps to study its monotonicity:

1. Find the derivative function $f'(x)$.

2. Determine the points where the derivative is equal to zero (critical points).

3. Place these points on the number line and analyse the sign of $f'(x)$ in each interval to show where the function is increasing and where it is decreasing.

The town planning department of a city is studying the noise level, $N(t)$ in decibels, on a central street from 8:00 ($t=0$) to 12:00 ($t=4$), according to the function:

$N(t) = t^2 - 4t + 70$

Use the first derivative to justify whether the noise increases or decreases at 9:00 ($t=1$) and at 11:00 ($t=3$). At what exact moment does the noise stop decreasing and start increasing?

The graph shown below is NOT the function $f(x)$, but the graph of its derivative $f'(x)$. Looking at it carefully, state with reasons the intervals of increase and decrease of the original function $f(x)$.

Remember: The growth of $f(x)$ depends on the sign of $f'(x)$.

Study the intervals of increase and decrease of the constant function $f(x) = 7$.

1. Calculate its derivative.

2. Based on the result, can you say that the function increases or decreases at any point? Justify your answer by relating it to the gradient of the tangent line.

A group of Year 13 students analyses the amount of material they remember from a History exam over time. The function modelling the remembered material is $M(d) = \frac{100}{d+1}$, where $d \ge 0$ is the number of days elapsed after the exam.

Calculate the derivative $M'(d)$ and prove, using its sign, that the function is decreasing for any value of $d$. Does this result make sense in a real-world context?

Given the function $f(x) = \frac{1}{2}x^2 - 4x + 1$, complete the following table of signs for the derivative and determine the intervals of increase and decrease.

State the value of the gradient of the tangent line at the point $x = 4$.

Lucía needs to study the increasing and decreasing intervals of the rational function $f(x) = \frac{2x-1}{x-2}$. After differentiating correctly, she obtains $f'(x) = \frac{-3}{(x-2)^2}$. Since the equation $f'(x) = 0$ has no real solution, Lucía concludes on her exam: «Since the derivative is never zero and its numerator is always negative while the denominator is always positive, the function has no distinct intervals of monotonicity and is strictly decreasing on all of $\mathbb{R}$».

1. a) Identify the conceptual mistake Lucía makes when determining the intervals of monotonicity and explain why her statement about all of $\mathbb{R}$ is false.
2. b) Correct her solution: state the domain of the function, divide the real line appropriately, and write down the correct decreasing intervals.
3. c) If the function decreases on each of its intervals, how do you explain that $f(1) = -1$ is less than $f(3) = 5$? Justify why this does not contradict the decreasing nature of the function on its intervals.

A traditional horchata stand in Valencia opens in the afternoon, between 11:00 and 20:00 ($t \in [11, 20]$). The number of customers served per hour is modeled by the quadratic function $C(t) = -t^2 + 32t - 216$.

1. a) Without calculating the derivative yet, reasonably estimate at what approximate time of the day (beginning, middle, or end) you would expect the customer influx to be increasing and when decreasing, taking into account the shape of the parabola.
2. b) Calculate the derivative function $C'(t)$, determine the critical point, and formally find the intervals of increase and decrease within the domain $[11, 20]$.
3. c) Compare the analytical result with your initial estimate and evaluate whether the behavior of the mathematical model is consistent with the usual customer flow at a shop throughout the afternoon.

A mobile app development company has found that its monthly profits, in thousands of euros, follow the function $B(x) = \frac{1}{3}x^3 - 4x^2 + 7x + 20$, where $x$ represents the time in months since the launch of its latest update ($0 \le x \le 12$). Determine the time intervals in which the company's profits are increasing and those in which they are decreasing.

The average manufacturing cost of a limited edition of designer hoodies is given by the function $C(x) = \frac{x^2 + 25}{x}$, where $x$ is the number of hoodies produced in hundreds of units and $C(x)$ is the cost in euros per unit. Knowing that production must be between 100 and 1 000 hoodies ($1 \le x \le 10$), study the intervals of increase and decrease of the average cost to help the designer decide how many hoodies to produce.

A streamer is analysing the evolution of their live viewers during a 6-hour broadcast. The function modelling the number of viewers (in thousands) is $E(t) = -t^3 + 9t^2 - 15t + 10$, where $t$ is the elapsed time in hours ($0 \le t \le 6$). Work out in which periods of the broadcast the number of viewers is increasing and in which it is decreasing.

Below you have the graph of the derivative function $f'(x)$ of a function $f(x)$ that is continuous on all of $\mathbb{R}$. The graph of $f'(x)$ is a parabola that crosses the $X$ axis at $x=-2$ and at $x=3$.

Identify the intervals where the original function $f(x)$ is increasing and decreasing. Justify your answer based on the relationship between the sign of the derivative and the monotonicity of the function.

The amount of carbon monoxide dissolved in the air near an industrial area after implementing a new filter is measured by the function $G(t) = t^2 - 12\ln(t+1)$, where $t$ is the time in days ($0 \le t \le 10$). Determine the time intervals in which the pollution is decreasing and when it starts to increase again.

We want to model the spread of a new viral challenge on a social network. The function describing the number of people who know about the challenge as a function of time $t$ (in days) is $P(t) = \frac{100t}{t^2 + 9}$. Analyse the monotonicity of this function for values of $t \ge 0$ and explain what happens to the growth of the challenge in the long term.

Find the value of the parameter $a$ such that the function $f(x) = x^3 + ax^2 + 24x + 1$ is increasing everywhere on its domain. For the value of $a = 9$, specifically determine the intervals of increase and decrease of the resulting function.

A forest surveillance drone follows a trajectory whose height in meters above ground level is given by $H(t) = t \cdot e^{-0,2t}$, where $t$ is the flight time in minutes ($0 \le t \le 20$). The figure shows the graph of $H(t)$. Using the derivative, study the time intervals in which the drone is gaining altitude and when it starts to descend.

The financial department of a transport company analyzes the average operating cost per trip as a function of the load $x$ (in tons), using the function $R(x) = \frac{x^2 + k}{x - 1}$, defined for $x \neq 1$.

1. a) Reasonably determine the value of the real parameter $k$ so that the function is strictly increasing on $(-\infty, -1) \cup (3, +\infty)$ and strictly decreasing on $(-1, 1) \cup (1, 3)$.
2. b) An analyst on the team states: «To guarantee those monotonicity intervals, it is sufficient to require that the derivative vanishes at $x = -1$». Justify analytically that this condition is necessary and study whether, for this family of functions, it is also sufficient.
3. c) Analyze whether there is any value of the parameter $k$ for which the function has no interval of decrease on its entire domain. Justify your answer by studying the sign of the numerator of $R'(x)$.

To study the intervals of increase and decrease of the function $f(x) = \frac{e^x}{x+2}$, two students propose different strategies after finding the derivative $f'(x) = \frac{e^x(x+1)}{(x+2)^2}$:

• Mateo proposes substituting numerical test values into the derivative (such as $x = -3$, $x = -1,5$, and $x = 0$) to deduce the sign of $f'$ in each region using a calculator.

• Sara proposes studying the sign by algebraically analyzing each factor of $f'(x)$: she argues that, within the domain, both $e^x$ and $(x+2)^2$ are strictly positive for all $x$, so the sign of $f'(x)$ depends exclusively on the factor $(x+1)$.

1. a) Compare both methods: evaluate the speed, mathematical precision, and risks of making mistakes associated with each.
2. b) Apply Sara's method to rigorously obtain the points that divide the real line and the intervals of increase and decrease of $f$.
3. c) Explain what serious mistake Mateo would have made if, carelessly, he had only considered as a partition point the value where $f'(x) = 0$ and had omitted the discontinuity at $x = -2$.

Consider the rational function $f(x) = \frac{x^2 + ax}{x - 4}$, where $a$ is a real parameter.

1. a) Determine the value of $a$ such that the function has a critical point at $x = 2$.
2. b) For the value of $a$ found previously, study the intervals of increase and decrease of the function, taking into account its domain.
3. c) Is there any value of $a$ for which the function is always increasing on the intervals $(-\infty, 4)$ and $(4, +\infty)$? Justify your answer analytically.

The accompanying graph shows the derivative function $f'(x)$ of a continuous function $f(x)$ defined on all of $\mathbb{R}$. Based on observing the graph of $f'(x)$, answer with reasons:

1. a) Determine the intervals of increase and decrease of the original function $f(x)$.
2. b) Identify the x-coordinates of the relative maximums and minimums of $f(x)$.
3. c) If we know that $f(0) = 0$, sketch the graph of the original function $f(x)$, paying special attention to its monotonicity.

A financial analyst has determined that the net profit of a tech company (in millions of euros) follows the function $B(t) = (t^2 - 8t + 15) \cdot e^{0,5t}$, where $t$ represents the time in years since its founding ($0 \le t \le 10$).

1. a) Calculate the marginal profit function, $B'(t)$.
2. b) Study the time intervals in which the company's profit is increasing and those in which it is decreasing.
3. c) In which exact year does the company move from increasing losses to decreasing losses? Interpret the result based on monotonicity.

Given the function $f(x) = \ln(x^2 - kx + 9)$, study its monotonicity depending on the values of the parameter $k$. In particular:

1. a) Determine the domain of the function for $k = 0$ and study its intervals of increase and decrease.
2. b) Find the values of $k$ for which the function has a single critical point.
3. c) If $k = 10$, explain why it makes no sense to study the increase across the entire real line $\mathbb{R}$.

Consider the piecewise-defined function:

1. a) Find the values of $a$ and $b$ such that the function is differentiable at $x = 0$.
2. b) For the values obtained, determine the intervals of increase and decrease of the function across its entire domain.
3. c) Determine whether there is any point where the function changes from increasing to decreasing without being a point where the derivative is equal to zero.

The number of active users on a new social network (in thousands) changes according to the function $N(t) = \frac{100t}{t^2 + 2t + 1}$, where $t \ge 0$ is the time in months.

1. a) Analyze the growth rate of the social network during the first year. At what exact moment does the number of users stop growing?

1. b) Prove that, from a certain month onward, the social network will enter an irreversible decline in terms of the number of users. State from which month this happens.

1. c) How does the monotonicity of the function behave as $t$ tends to infinity? Relate this fact to the existence of a horizontal asymptote.

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

1. a) Determine its domain and study the sign of the function.
2. b) Calculate the intervals of increase and decrease. Pay special attention to the behaviour of the function near $x = 0$.
3. c) Find the equation of the tangent line to the curve at the point where the function changes from decreasing to increasing.

A renewable energy company estimates that the marginal cost of producing $x$ megawatt-hours is given by the function $C'(x) = \frac{x-5}{\sqrt{x^2+11}}$.

1. a) Without calculating the total cost function $C(x)$, explain in which production intervals the total cost is an increasing function and in which it is decreasing.
2. b) For what production quantity is the minimum cost achieved? Justify your answer using the study of the monotonicity of the cost function.
3. c) If the company decides that it is only profitable to produce in the interval where the cost is increasing, what is the minimum production it should consider?

A Spanish financial institution markets a savings product whose effective annualized return $R(t)$ (expressed as a percentage) depends on the investment period $t$ (measured in years, with $t > 0$), following the model:

$R(t) = \frac{2 + \ln(t)}{t}$.

1. a) Determine the mathematical domain of validity of the function and explain the reason why the real financial context prevents evaluating the return at $t = 0$.
2. b) Calculate the derivative $R'(t)$, determine its critical points, and deduce the intervals of increase and decrease of the return, justifying the sign of the derivative on each interval. Express the endpoints of the intervals both as exact values and with an approximation to two decimal places.
3. c) Advise an investor by justifying your answers: for what period of time is it advisable for them to keep the money deposited if they want their annualized rate to continue increasing? If a client leaves the funds deposited over a very long term ($t \to +\infty$), toward what value does the return tend?

Consider the family of functions $f(x) = (x^2 + ax + b)e^{-x}$, where $a$ and $b$ are real parameters.

1. a) Find the values of $a$ and $b$ knowing that the function has critical points at $x = 0$ and at $x = 3$.
2. b) For the values of $a$ and $b$ found in the previous section, determine in detail the intervals of increase and decrease of $f$. Justify the sign of $f'(x)$ on each interval based on the sign of the resulting polynomial and the properties of the exponential function.
3. c) Keeping fixed the value of $a$ obtained in section a), investigate whether it is possible to choose the parameter $b$ such that the function is monotonically decreasing on all of $\mathbb{R}$. If it is possible, determine for which values of $b$ it occurs by analyzing the discriminant of the equation associated with the derivative; if it is not, prove its impossibility.