Intervals of Increase and Decrease

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.

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.

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 $x = 4$.

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)?

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 $f'(x)$.

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 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 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?

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?

A small phone case design company has determined that its daily profits, in pesos, 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?

A mobile app development company has found that its monthly profits, in thousands of pesos, 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.

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.

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 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?

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 pesos 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.

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.

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 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.

A forest surveillance drone follows a trajectory whose height in meters above ground level is given by $H(t) = t \times e^{-0.2t}$, where $t$ is the flight time in minutes ($0 \le t \le 20$). Study the time intervals in which the drone is gaining height and when it starts to descend.

A financial analyst has determined that the net profit of a tech company (in millions of pesos) follows the function $B(t) = (t^2 - 8t + 15) \times e^{0.5t}$, 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.

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.

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

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 behavior 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.

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.