Growth and Extrema with the Derivative

Determine the intervals where the function $f(x) = x^3 - 3x$ is increasing and decreasing by studying the sign of its derivative.

The figure shows the graph of the derivative $f'$ of a function $f$, which intersects the x-axis at $x = -1$ and at $x = 3$. By observing the sign of $f'$ in the figure, state the intervals where $f$ is increasing and decreasing and at which points it has local extrema.

A student studies the monotonicity of $f(x) = x^3 - 3x^2$ and writes: "$f'(x) = 3x^2 - 6x$, which is equal to zero at $x = 0$ and at $x = 2$; therefore, $f$ is increasing on $(0, 2)$ and decreasing elsewhere." Identify the error made in the conclusion, correct it by carefully studying the sign of $f'$, and write down the correct intervals of monotonicity.

Find the local extremum of the function $f(x) = x^2 - 6x + 5$ using the derivative, and determine with reasons whether it is a maximum or a minimum.

Consider the function $f(x) = x^3 - 3x^2 - 9x + 2$.

1. a) Calculate its derivative and find its stationary points.

1. b) Determine the intervals where the function is increasing and decreasing.

1. c) Classify each stationary point as a local maximum or local minimum and state the value of the function at these points.

The function $f(x) = x^3 + ax^2 + bx$ has a local extremum at the point $(1, -2)$.

1. a) Set up the two conditions that $a$ and $b$ must satisfy and calculate their values.

1. b) Using these values, locate the other local extremum of the function and classify both.

Calculate the stationary points of the function $f(x) = x^3 - 12x + 1$, that is, the points where its derivative is zero.

The figure shows the graph of a function $f$ defined on $[-3, 4]$, with its two local extrema indicated. Reading from the graph:

1. a) State the intervals where $f$ is increasing and decreasing.

1. b) Estimate the coordinates of the local maximum and local minimum.

1. c) Also state the absolute maximum and absolute minimum of $f$ on $[-3, 4]$.

Let $f(x) = x^3 - 3x + 1$ be defined on the closed interval $[-2, 3]$.

1. a) Locate its stationary points within the interval.

1. b) Calculate the value of $f$ at the stationary points and at the endpoints of the interval.

1. c) Determine the absolute maximum and minimum of $f$ on $[-2, 3]$ and explain why it is not sufficient to check only the stationary points.

After administering a medication, its blood concentration, in $\text{mg/l}$, follows the model $C(t) = 10t \times e^{-0.5t}$, where $t \geq 0$ is measured in hours, as shown in the figure.

1. a) Calculate $C'(t)$.

1. b) Determine the time at which the concentration is at a maximum.

1. c) Determine the intervals in which the concentration increases and those in which it decreases.

Fully analyse the monotonicity of the function $f(x) = \frac{x}{x^2 + 1}$: increasing and decreasing intervals, local extrema with their values, and behavior as $x \to \pm \infty$.

Study the monotonicity and extrema of the function $f(x) = x \times e^{-x}$ on all of $\mathbb{R}$, indicating the value of the local extremum and the limit of the function as $x \to +\infty$.

At a fair, a water rocket reaches a height of $h(t) = -4t^2 + 24t + 1$ meters $t$ seconds after launch. Using the derivative, calculate the time at which it reaches its maximum height and find this maximum height.

The organizers of a festival estimate that if they sell tickets at $x$ pesos each, $400 - 8x$ people will attend, so that the revenue is $I(x) = x \times (400 - 8x)$ pesos, with $0 \leq x \leq 50$.

1. a) Determine the intervals in which revenue increases and those in which it decreases as the price rises.

1. b) Find the price that maximizes the revenue and the value of this maximum revenue.

The air quality index of a city during a pollution episode day is modelled, over the period $0 \leq t \leq 24$ hours, by $I(t) = t^3 - 27t^2 + 216t + 50$.

1. a) Find the times at which the trend of the index changes.

1. b) Determine the time intervals during which the index worsens (increases) and those during which it improves (decreases).

1. c) Determine the maximum value of the index between $0$ and $14$ hours and at what time it is reached.

1. d) Reason whether the model would still be valid for predicting the index several days later.