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.

Aoife is studying the extrema of the function $f(x) = (x - 2)^3 + 5$. To do so, she computes its derivative $f'(x) = 3(x - 2)^2$, sets it to zero, and obtains the solution $x = 2$. She immediately concludes in her notebook: "Since the derivative vanishes at $x = 2$, the function has a relative extremum (minimum or maximum) at that point".

1. a) Explain precisely what conceptual error Aoife makes when deducing the existence of an extremum based solely on $f'(2) = 0$.
2. b) Correct her resolution: study the sign of $f'(x)$ to the left and to the right of $x = 2$ and determine what actually happens to the growth of the function in that neighborhood.
3. c) Give a verbal or everyday-life example that helps her understand the difference between stopping momentarily with zero velocity (horizontal tangent) and turning around (change of direction).

Given the polynomial function $f(x) = 2x^3 - 9x^2 + 12x$ restricted to the closed interval $[0, 3]$:

1. a) Before calculating any derivative, evaluate mentally or quickly the endpoints of the interval, $f(0)$ and $f(3)$, as well as the value at $f(1)$. From these three data points, make a reasoned estimate of whether you think the absolute maximum will be reached in the interior of the interval or at the right endpoint $x = 3$.
2. b) Calculate the derivative function $f'(x)$, determine the stationary points that belong to the interval $[0, 3]$, and classify them by studying the sign of $f'$.
3. c) Find the exact absolute maximum and minimum on $[0, 3]$ by evaluating the relevant candidates (interior stationary points and interval endpoints) and check whether your initial estimate was correct.

At a fair, a water rocket reaches a height of $h(t) = -4t^2 + 24t + 1$ metres $t$ 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 euros, is given by $C(x) = x + \frac{1600}{x}$, with $x > 0$. Find the number of bicycles that minimises 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 organisers of a festival estimate that if they sell tickets at $x$ euros each, $400 - 8x$ people will attend, so that the revenue is $I(x) = x \times (400 - 8x)$ euros, 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 maximises the revenue and the value of this maximum revenue.

Met Éireann records the temperature evolution during a summer day at a station in the Irish midlands (Birr, Co. Offaly). The temperature $T$ (in degrees Celsius) throughout the $24$ hours of the day is modeled by the function:

where $t$ represents the time elapsed in hours since midnight ($t = 0$).

1. a) Calculate the derivative $T'(t)$ and determine the intervals of the day during which the temperature was increasing and during which it was decreasing.
2. b) Find at what exact time the relative (local) minimum temperature of the early morning was recorded and at what time the absolute maximum was reached, calculating the thermal value at both moments.
3. c) Interpret the sign of the derivative $T'(t)$ in the period from $14:00$ to $16:00$ hours to explain why the daily maximum temperature does not coincide with solar noon (the moment of maximum insolation), but instead exhibits thermal lag.

To classify the stationary points of a differentiable function, one can study the sign change of the first derivative or resort to the sign of the second derivative.

1. a) Consider the function $f(x) = x^4 - 4x^3 + 1$. Find its two stationary points and try to classify both using the second derivative $f''(x)$. What insurmountable obstacle arises at $x = 0$? Justify which method must necessarily be applied at that point and conclude what type of point it is.
2. b) Now consider the rational function $g(x) = \frac{x}{x^2 + 4}$. Find its stationary points. Compare the algebraic work required to compute $g''(x)$ versus studying the sign of $g'(x)$ taking advantage of the fact that the denominator of $g'$ is always strictly positive. Justify which of the two approaches is more convenient and classify the relative extrema of $g$.

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

In a skatepark, there is a plan to build a ramp that smoothly connects two horizontal sections located at different heights. In a coordinate system where distances are measured in meters, a third-degree polynomial function $f(x) = ax^3 + bx^2 + cx + d$, defined on the interval $[0, 4]$, is sought that simultaneously satisfies:

1. At $x = 0$, the ramp starts at an elevation of $3\text{ m}$ with a horizontal tangent.

2. At $x = 4$, the ramp reaches ground level ($0\text{ m}$) also with a horizontal tangent.

3. The ramp must be strictly decreasing throughout the interior $(0, 4)$ so that no skater encounters counter-slopes or intermediate crests.

1. a) State the four conditions on $f$ and $f'$ necessary to calculate the coefficients $a$, $b$, $c$, and $d$, and obtain the algebraic expression of $f(x)$.
2. b) Analytically prove that the calculated function satisfies the condition of strict decrease in $(0, 4)$ by studying the sign of its derivative.
3. c) Explain in detail why it would have been mathematically impossible to solve this problem using a second-degree polynomial instead of a third-degree one.

Consider the family of functions $f_k(x) = x^3 - 3kx^2 + 9x - 1$, where $k \in \mathbb{R}$ is a parameter.

1. a) Calculate $f'_k(x)$ and determine for which values of $k$ the function is strictly increasing on all of $\mathbb{R}$. Explain what happens to the graph and the tangent line in the particular case $k = \sqrt{3}$.
2. b) Find the values of $k$ for which the function has exactly two distinct stationary points. Classify them as a relative maximum or minimum, justifying your answer according to the signs adopted by the derivative along the real line.
3. c) For the specific case $k = 2$, restrict the function to the closed interval $[0, 5]$. Determine with reasoning all candidates for absolute extrema, evaluate the function at them, and locate the absolute maximum and minimum on that interval. In light of the result, argue why on a closed interval the absolute maximum does not necessarily have to coincide with the relative maximum.