Exercises: Monotonicity and extrema of a function

Calculate the Average Rate of Change (ARC) of the function $f(x) = x^2 - 4x + 3$ on the interval $[2, 5]$. From the result, can you deduce whether the function is, on average, increasing or decreasing on that interval?

Observe the following graph of a polynomial function. Identify the coordinates of the points $A$ and $B$, and classify them as local maxima or local minima. Then, write the intervals where the function is strictly increasing and where it is strictly decreasing.

Find analytically the vertex of the quadratic function $f(x) = 3x^2 - 12x + 7$. State whether this point corresponds to an absolute maximum or an absolute minimum of the function.

Using the algebraic definition of monotonicity (evaluating $f(x)$ and $f(y)$ for two arbitrary values $x < y$), prove that the linear function $f(x) = 5x - 3$ is strictly increasing throughout its domain $\mathbb{R}$.

The following graph represents a quadratic function. Determine the coordinates of its vertex $V$. Is it a maximum or a minimum? Is it local, absolute, or both?

Calculate the Average Rate of Change of the rational function $f(x) = \frac{4}{x}$ on the interval $[1, 4]$. What does the sign of the result tell us about the behaviour of the function on that interval?

A stone is thrown vertically upwards. The height in metres it reaches as a function of time in seconds is given by the expression $h(t) = -5t^2 + 30t + 2$. Calculate the time at which the stone reaches its maximum height and determine what this height is.

Check whether the point $x = 3$ is a local minimum for the function $f(x) = x^2 - 6x + 10$. To do this, evaluate the function at the point itself and at two values very close to its left and right, for instance $x = 2,9$ and $x = 3,1$.

A surveillance drone ascends and descends vertically. Its height in metres after $t$ seconds is given by the function $h(t) = -t^2 + 4t + 5$ during the interval $t \in [0, 4]$. Mateo calculates $h(0) = 5\text{ m}$ and $h(4) = 5\text{ m}$, and from there he obtains $\text{TVM}[0, 4] = \dfrac{5 - 5}{4 - 0} = 0\text{ m/s}$. With this result, Mateo states: "Since the average rate of change is 0, the drone has remained still at a height of $5\text{ m}$ the entire time and the function is constant, so it has no maxima or minima in that interval". Identify Mateo's conceptual error, explain what the secant line geometrically represents in this case, and find the coordinates of the vertex of the parabola to demonstrate with reasoning what actually happened to the drone's height.

From AEMET records for a clear spring day in Seville, the temperature $T(t)$ in degrees Celsius between 8:00 and 16:00 ($t \in [8, 16]$) is modelled by the function $T(t) = -0,5t^2 + 14t - 60$.

1. a) Before carrying out detailed numerical calculations, look at the coefficients of the parabola and mentally calculate the time of the vertex. Based on that information, estimate whether $\text{TVM}[8, 12]$ must be positive or negative, and whether you expect the average warming to be greater or less than $5\text{ }^\circ\text{C/h}$. Justify your prior estimation.
2. b) Calculate $\text{TVM}[8, 12]$ accurately by applying its definition. Compare the obtained value with your prior estimation and write a conclusion interpreting the physical meaning of the result for the city's inhabitants.

In a laboratory culture, the number of bacteria $N(t)$ as a function of time $t$ (in hours) follows the law $N(t) = 20t^2 + 100t + 500$. Calculate the average rate of change in the interval $[1, 4]$ and interpret the meaning of the result in the context of the problem.

The following graph shows a piecewise function. From it, determine the intervals where the function is increasing and decreasing. Also identify the coordinates of the absolute minimum and state whether any maximum exists. Each grid square represents $1$ unit on both axes.

Find the value of the parameter $k$ so that the quadratic function $f(x) = -x^2 + kx + 5$ reaches its maximum value precisely at the x-coordinate $x = 4$. Once $k$ is found, calculate this maximum value.

The profit of an electronic components company, expressed in thousands of euros, is given by the function $B(x) = -2x^2 + 80x - 300$, where $x$ is the number of batches produced. How many batches should the company produce to maximise its profit? How much is this maximum profit?

Prove algebraically that the function $f(x) = 2x^3 + 5$ is strictly increasing for all $x \in \mathbb{R}$. To do so, take any two values $x_1$ and $x_2$ such that $x_1 < x_2$ and check that $f(x_1) < f(x_2)$.

The temperature inside a chemical reactor during the first hours of a process follows the function $T(t) = -0,2t^2 + 2,4t + 20$, with $t$ in hours and $T$ in degrees Celsius. Determine the maximum temperature reached and the time interval during which the reactor temperature is decreasing.

Calculate and compare the average rate of change of the functions $f(x) = 3^x$ and $g(x) = 2x^2$ in the interval $[1, 3]$. Which of the two functions experiences greater average growth in that interval?

Determine the intervals of monotonicity (increasing and decreasing) of the rational function $f(x) = \frac{-3}{x-2}$. You can justify your answer by analysing the average rate of change in generic intervals or using the graphical properties of hyperbolas.

The specific electrical consumption of a high-speed train unit in Spain, expressed in kWh per kilometre travelled, depends on its speed $v$ (measured in hundreds of $\text{km/h}$, for the operational domain $v \in [1, 3]$) according to the quadratic function $E(v) = 4v^2 - 12v + 25$.

1. a) Justify with reasoning why this function has an absolute minimum and not a maximum in its domain, and calculate at what exact speed (in $\text{km/h}$) the consumption per kilometre is minimised, as well as this minimum consumption.
2. b) Determine whether the function is bounded in the closed interval $[1, 3]$, indicating with justification its optimal lower bound and its optimal upper bound.
3. c) Calculate $\text{TVM}[1; 1,5]$ and $\text{TVM}[2; 2,5]$. Explain, considering the sign and magnitude of both rates, how the train's efficiency varies when travelling below or above its optimal speed.

In a school track and field competition, the trajectory of a shot put is recorded using a quadratic function $f(x) = ax^2 + bx + c$, where $x$ is the horizontal distance travelled in metres and $f(x)$ is the height in metres. The following data are known:

1. The shot leaves the athlete's hand at a height of $2\text{ m}$ (that is, $f(0) = 2$).

2. In the first horizontal section of two metres, the average rate of change is $\text{TVM}[0, 2] = 0,6$.

3. Between two and six metres of horizontal distance, the average rate of change is $\text{TVM}[2, 6] = 0$.

1. a) Set up the corresponding system of equations and determine the coefficients $a$, $b$, and $c$ with reasoning.
2. b) Find the horizontal position $x$ at which the shot reaches its maximum height and calculate this height, justifying why it is an absolute maximum of the throw.
3. c) From the obtained vertex, specify the intervals where the function is strictly increasing and strictly decreasing.

A farmer has $120$ metres of fencing to enclose a rectangular vegetable garden. One side of the garden will rest against the straight wall of a barn, so no fencing is needed on that side. Find the dimensions of the garden that maximise its area and calculate this area.

A quadratic function of the form $f(x) = ax^2 + bx + c$ is known to reach a relative maximum at the point $(3, 10)$ and cross the y-axis at the point $(0, 1)$. Determine the values of the parameters $a$, $b$ and $c$. Subsequently, write the intervals where the function is strictly increasing and strictly decreasing.

Show that the function $f(x) = \frac{8}{x^2+4}$ has an absolute maximum at $x=0$. To achieve this, prove algebraically that the inequality $f(x) \le f(0)$ holds for any real number $x$.

A factory sells a certain product at a unit price that depends on demand, given by the function $p(x) = 200 - 0,5x$ (in euros), where $x$ is the quantity of units sold. The total production costs follow the function $C(x) = 50x + 2000$. Find the profit function $B(x)$, determine the production level that maximises profit and calculate the maximum profit.

The following graph represents a rational function with a vertical asymptote and an oblique asymptote. Observing the graph, identify the approximate coordinates of its relative extrema (local maximum and local minimum). Then, write the intervals in which the function is increasing and decreasing.

Given the piecewise function:

$f(x) = \begin{cases} -x^2 + 6x - 5 & \text{if } x \le 4 \\ 2x - 5 & \text{if } x > 4 \end{cases}$

Study its intervals of monotonicity analytically and find the coordinates of its relative or absolute extrema. Justify whether the boundary point between branches ($x=4$) represents any type of extremum.

The concentration of a drug in the bloodstream, measured in mg/L, as a function of the hours elapsed since its administration, is modelled by the function $C(t) = \frac{12t}{t^2+4}$ for $t \ge 0$. Observe the attached graph. Estimate the time at which the concentration is maximum. Then, calculate the average rate of change in the intervals $[0, 2]$ and $[2, 6]$ to confirm the change in monotonicity analytically.

Prove that the quadratic function $f(x) = -3x^2 + 18x - 15$ is bounded above. Calculate its absolute maximum and determine exactly the interval in which the function is strictly decreasing.

Given the rational function $f(x) = \dfrac{3x + 1}{x + 2}$, defined on the interval $I = (-2, +\infty)$, the goal is to prove that it is strictly increasing on its entire domain. Two different analytical procedures are proposed:

• Method 1 (Direct difference): Take two generic values $x_1, x_2 \in I$ with $x_1 < x_2$, calculate the difference $f(x_2) - f(x_1)$, transform it into a single simplified fraction, and study algebraically the resulting sign of the numerator and the denominator.

• Method 2 (Generic Average Rate of Change): Form the expression $\text{TVM}[x_1, x_2] = \dfrac{f(x_2) - f(x_1)}{x_2 - x_1}$ for $x_1 < x_2$, simplify the fraction algebraically by dividing by $(x_2 - x_1)$, and determine the sign of the quotient.

1. a) Apply both methods rigorously, developing all algebraic steps.
2. b) Compare the two approaches: which one connects more immediately with the slope of the secant line? Does either of them offer algebraic advantages over the other, or are they conceptually equivalent? Justify your answer.
3. c) Based on the proven monotonicity, reason whether $f$ can have relative extrema in $(-2, +\infty)$ and prove whether the function is bounded above on the interval $[0, +\infty)$, indicating an upper bound if one exists.

The altitude profile of a time-trial stage in Sierra Nevada is designed using a quadratic function $h(x) = ax^2 + bx + c$, where $x$ is the distance travelled in kilometres ($x \in [0, 10]$) and $h(x)$ is the altitude in hundreds of metres. The course must meet three requirements:

1. The start ($x = 0$) is located at an altitude of $800\text{ m}$ ($h(0) = 8$).

2. In the first four kilometres, the average uphill slope is $200\text{ m/km}$ (that is, $\text{TVM}[0, 4] = 2$).

3. The upper bound of the altitude throughout the stage is reached exactly at kilometre $x = 6$, this point being the summit of the route.

1. a) Deduce the formula for $h(x)$ with reasoning. Justify why the coefficient $a$ must be negative based on the condition that an upper bound is attained.
2. b) Prove, without using derivatives and solely employing the formal definition of monotonicity ($x_1 < x_2 \Rightarrow h(x_1) > h(x_2)$), that the route is strictly decreasing in the final section $[6, 10]$. To do this, factor the difference $h(x_2) - h(x_1)$ and analyse the sign of each factor.
3. c) Determine with reasoning whether the function is bounded on the interval $[0, 10]$ and find the absolute lower and upper bounds reached during the race, indicating at which kilometres they occur.