Intervals of increase and 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?

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?

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.

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.

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 $\mathbb{R}$.