Exercises: Increasing and decreasing intervals and extrema using derivatives

It is known that the derivative of a function $f$ satisfies $f'(3) = -2$. Without knowing the expression for $f$, determine with reasons whether the function is increasing or decreasing at the point $x = 3$ and explain what this value means for the tangent line at that point.

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

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.

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

The derivative of a function is $f'(x) = (x - 1) \times (x - 4)$. By studying the sign of each factor on the number line, determine the intervals where $f$ is increasing and decreasing, and locate its local extrema.

Determine the intervals of monotonicity of the function $f(x) = 2x^3 - 9x^2 + 12x - 3$.

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.

State, giving reasons, whether the following statement is true or false: "If $f'(a) = 0$, then $f$ has a local maximum or minimum at $x = a$." The figure, which shows the function $f(x) = x^3$ and its tangent at the origin, may help you decide.

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.

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.

A small workshop manufactures $x$ bicycles per month, and the average cost per bicycle, in pesos, is given by $C(x) = x + \frac{1600}{x}$, with $x > 0$. Find the number of bicycles that minimizes the average cost and the value of this minimum cost, confirming with the sign of the derivative that it is indeed a minimum.

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.

Find the value of the parameter $b$ such that the function $f(x) = x^2 + bx + 3$ has a local minimum at the point $x = 2$. With this value of $b$, calculate the minimum value of the function.

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.

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 $[-3, 4]$.

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.

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 $x \to \pm \infty$.

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.

Consider the function $f(x) = x^3 + x - 1$.

1. a) Show, by studying the sign of $f'(x)$, that $f$ is strictly increasing throughout $\mathbb{R}$.

1. b) Using the previous part and the values $f(0)$ and $f(1)$, determine with reasons how many real solutions the equation $x^3 + x - 1 = 0$ has.

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.

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.

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 $x \to +\infty$.

We want to prove that $e^x \geq 1 + x$ for all $x \geq 0$ using monotonicity.

1. a) Define $f(x) = e^x - 1 - x$ and study the sign of $f'(x)$ for $x \geq 0$.

1. b) Calculate $f(0)$ and deduce, from the monotonicity of $f$, that $f(x) \geq 0$ for all $x \geq 0$.

1. c) Conclude the required inequality and reason whether it is also valid for $x < 0$.

The figure shows the graph of the derivative $f'$ of a certain function $f$ differentiable on all of $\mathbb{R}$; the curve crosses the x-axis at exactly $x = -2$, $x = 1$ and $x = 4$.

1. a) State the points at which $f$ has local extrema and classify them by studying the sign of $f'$.

1. b) Is it possible to know the value that $f$ takes at those extrema from the graph of $f'$? Justify your answer.