Exercises: Increasing, decreasing, maximums and minimums

Calculate the average rate of change of the function $f(x) = 4x - 7$ in the intervals $[-2, 1]$ and $[1, 5]$. In light of the results, what relationship exists between the average rate of change and the gradient of a linear function?

Given the quadratic function $g(x) = x^2 - 6x + 8$, find its average rate of change in the interval $[2, 4]$. What does the obtained result indicate about the value of the function at the endpoints of the interval?

Observe the graph of the function $f(x)$ and determine its intervals of increase and decrease. Also write the coordinates of its relative maximums and minimums.

The following table shows the daily profits (in hundreds of euros) of a small shop over one week:

Identify on which days the relative maximums and minimums of the profits are reached during that week, justifying your answer based on increase and decrease.

Find the coordinates of the vertex of the parabola with equation $y = -3x^2 + 12x - 5$. State whether that vertex corresponds to an absolute maximum or an absolute minimum of the function.

Without plotting its graph, determine the intervals of increase and decrease of the quadratic function $f(x) = 2x^2 + 8x - 3$.

The temperature $T$ (in $^\circ\text{C}$) of a patient from admission to A&E is given by the formula $T(t) = 38,5 + 0,2t - 0,1t^2$, where $t$ is the time elapsed in hours. Calculate the average rate of change of the temperature in the first two hours ($t \in [0, 2]$).

If we know that the average rate of change of a continuous function in the interval $[1, 4]$ is negative, can we state with absolute certainty that the function is strictly decreasing throughout that interval? Reason your answer theoretically or using a graphical example.

Marcos is analyzing the performance of a municipal lawn mower over an $8$-hour workday. The amount of fuel consumed per hour (in liters) is modeled by the function $C(t) = -t^2 + 8t + 5$, where $t$ is the elapsed time in hours since the start of the workday ($t \in [0, 8]$). When studying the interval $[2, 6]$, Marcos calculates:

From this result, he concludes in his report: "Since the average rate of change is zero, the hourly fuel consumption did not vary at all between hour $2$ and hour $6$, having remained strictly constant at $17\text{ l/h}$ during those four hours."

1. a) Identify the conceptual error made by Marcos when interpreting the ARC.
2. b) Calculate the consumption at the intermediate hour $t = 4$ and use it to numerically refute his statement.
3. c) Explain what it actually means for the ARC to be equal to $0$ on that interval and what really happened to the consumption between $t = 2$ and $t = 6$.

The State Meteorological Agency (AEMET) records the evolution of temperature $T$ (in $^{\circ}\text{C}$) at a weather station in Córdoba during a hot May day, between $8:00$ ($t = 0$) and $20:00$ ($t = 12$). This evolution fits the quadratic function:

1. a) Calculate the average rate of change of the temperature during the morning period $[0, 4]$ (from $8:00$ to $12:00$) and during the afternoon period $[8, 12]$ (from $16:00$ to $20:00$). State the units and interpret the meaning of the sign obtained in each period.
2. b) Determine at what time of day the maximum temperature was reached and what that value was.

A projectile is launched vertically upwards such that its height in metres is given by the function $h(t) = -5t^2 + 40t + 10$, where $t$ is the time in seconds. Calculate the average rate of change of the height in the intervals $[0, 2]$ and $[6, 8]$. Physically interpret the meaning and the sign of the results obtained.

A bicycle factory estimates that its monthly profits, in thousands of euros, follow the function $B(x) = -0,5x^2 + 50x - 200$, where $x$ is the quantity of bicycles produced in tens of units. How many bicycles must they manufacture to obtain the maximum possible profit? How much will this profit be?

The attached graph shows the concentration of a medication in a patient's bloodstream (in mg/L) as a function of the time elapsed (in hours) since it was administered. Determine the intervals where the concentration increases and decreases, and state the time at which the maximum concentration is reached and its approximate value.

Find the values of the parameters $a$ and $b$ so that the quadratic function $f(x) = ax^2 + bx + 3$ reaches a relative maximum at the point $(2, 11)$.

The average production cost of $x$ decametres of a special cable is given by the function $C(x) = \frac{400}{x} + 5x$ (in euros). Calculate the average rate of change of the cost between $x = 10$ and $x = 20$. Does the average cost increase or decrease in that production interval?

Analytically study the increase, decrease and relative extrema of the following piecewise function:

$f(x) = \begin{cases} 2x + 6 & \text{if } x \le -1 \\ -x^2 + 2x + 7 & \text{if } x > -1 \end{cases}$

The number of users connected to a new social network grows according to the function $U(t) = 100 \cdot 2^t$, where $t$ is the time in months. Calculate the average rate of change in the intervals $[0, 2]$ and $[2, 4]$. Compare the results and explain what happens to the growth rate of the social network as time passes.

The elevation profile of a mountain stage in a cycle race is approximated by the attached graph. Identify at what distance from the start the mountain passes (relative maximums) and valleys (relative minimums) are located, stating the approximate altitude in each case.

During the passage of an Atlantic storm through Galicia, the volume of water $V(t)$ (in cubic hectometers, $\text{hm}^3$) accumulated in a reservoir is monitored over $6$ days ($t \in [0, 6]$). The evolution corresponds to the function:

1. a) Estimate: Without calculating the average rate of change yet, observe the concavity of the parabola and state, with reasoning, whether you expect the ARC to be greater in the initial interval $[0, 2]$ or in the intermediate interval $[2, 4]$. Do you expect both rates to be positive, or one of them to be negative? Justify your qualitative prediction.
2. b) Check: Algebraically calculate the ARC on $[0, 2]$ and on $[2, 4]$ with their corresponding units and confirm whether your estimate was correct.
3. c) Find on which day ($t$) the reservoir reaches its maximum level during this event and how many $\text{hm}^3$ it stores. From that day on, does the stored water increase or decrease? Justify your answer using the theoretical graph of the function.

A small cooperative oil mill in Jaén calculates the average processing cost $C(x)$ per liter of extra virgin olive oil (in euros) as a function of the total volume produced $x$ (expressed in thousands of liters, with $x \in [1, 10]$). The model corresponds to a parabola of the form:

where $a$ and $b$ are unknown coefficients.

From the previous season, two facts are known:

1) The average rate of change of the cost on the interval $[1, 3]$ was $-1,20$ € per additional thousand liters produced.

2) The minimum processing cost per liter was reached exactly when the mill produced $4000$ liters ($x = 4$).

1. a) From the abscissa of the minimum, write a mathematical relationship between the coefficients $a$ and $b$.
2. b) Using the algebraic definition of the ARC on $[1, 3]$, set up an equation and determine the exact numerical values of $a$ and $b$.
3. c) Write the complete function and calculate the minimum achievable cost per liter, as well as the cost per liter if $10\,000$ liters were produced ($x = 10$).

A farmer has $240$ metres of fencing to enclose a rectangular field adjacent to a straight river. If the side bordering the river does not need fencing, express the area of the field as a function of its width $x$. What should the dimensions of the field be so that the enclosed area is as large as possible?

A bus company charges $30$ euros per ticket and, at that price, manages to sell $400$ tickets daily. A market study indicates that for each euro the ticket price increases, $10$ fewer tickets will be sold. Express the company's daily revenue as a function of the price increase in euros, $x$. What ticket price maximises revenue and how many tickets will be sold at that price?

A window is to be constructed in the shape of a rectangle surmounted by a semicircle. If the total perimeter of the window (including the base, vertical sides and semicircular arc) must be exactly $8$ metres, determine the width of the base that allows the area of the window to be maximum, thus allowing the maximum amount of light to enter.

The average rate of change of a quadratic function $f(x) = ax^2 + bx + c$ in the interval $[-1, 1]$ is $2$, and the same type of calculation for the interval $[1, 3]$ gives $10$. Knowing that the function passes through the point $(0, -3)$, determine its complete algebraic expression and find the exact coordinates of its vertex.

The graph shows the growth in height of two plant species, A and B, over $8$ weeks in a greenhouse. By observing the curves, determine in which time interval plant A grew faster than plant B (comparing their average rates of change) and in which week the difference in height between them was zero.

The total daily production cost of $x$ decametres of a special fabric in a factory is given by the function $C(x) = 2x^2 - 120x + 2500$ (in euros). On the other hand, the daily revenue obtained from selling those $x$ decametres follows the function $I(x) = 80x$ (in euros). Write the function representing the daily profit $B(x)$ and determine how many decametres must be manufactured and sold for profit to be maximum. How much will this maximum profit be?

Given the function $f(x) = \frac{k}{x^2 + 4}$, determine the value of the parameter $k$ knowing that the absolute maximum value of the function is $5$. For that value of $k$, calculate the average rate of change in the interval $[0, 2]$ and in the interval $[-2, 0]$. What geometric property of the function justifies that the results have opposite signs?

Prove algebraically, by calculating the average rate of change in a generic interval $[x_1, x_2]$ (where $x_1 < x_2 < 0$), that the inverse proportion function $f(x) = -\frac{4}{x}$ is strictly increasing in the interval $(-\infty, 0)$.

An express parcel delivery company in Zaragoza studies the fuel expense $G(v)$ (in euros per $100\text{ km}$) of its delivery vans depending on their average driving speed $v$ (in $\text{km/h}$). The consumption follows the model:

Due to traffic limits and the type of road on the assigned route, the average speed can only vary within the closed interval $[30, 50]\text{ km/h}$.

To determine at what average speed the minimum expense and maximum expense are obtained, two analysts propose different strategies:

• Strategy A (Hugo): "Since the parabola opens upwards ($a > 0$), the minimum expense is always located at the vertex $v_v = -\dfrac{b}{2a}$. We calculate the vertex for the minimum, and for the maximum we evaluate the endpoints $v = 30$ and $v = 50$, keeping the larger one."

• Strategy B (Sara): "First, we must check whether or not the vertex belongs to the permitted interval $[30, 50]$. If it lies outside, the function does not change its direction of increase/decrease along our route, so both the absolute minimum and absolute maximum on the interval will necessarily be at the endpoints $v = 30$ and $v = 50$."

1. a) Calculate the abscissa of the vertex and analyze which of the two analysts is right, explaining the other's error with mathematical rigor.
2. b) Justify the correct general method for optimizing any quadratic function when its domain is restricted to a closed interval $[a, b]$.
3. c) Apply the correct method to find the speed of minimum expense, the speed of maximum expense, and the corresponding expense in euros for each within the permitted interval.

In a single-family home in Toledo with solar panels for self-consumption, the instantaneous power generated $P(t)$ (in kilowatts, $\text{kW}$) during a clear spring day is modeled by the function:

where $t$ represents the solar hour of the day, for the daytime interval $t \in [8, 18]$.

During those same hours, the home maintains a constant baseline electricity consumption of $C = 1,6\text{ kW}$. The net surplus power fed into the power grid for compensation is given by $E(t) = P(t) - C$.

1. a) Write the simplified algebraic expression of the surplus function $E(t)$ and calculate, with reasoning, the time interval during which the home feeds surplus energy into the grid (that is, when $E(t) \ge 0$).
2. b) Determine at what solar hour the maximum feed-in to the grid is reached and the value of this peak power in $\text{kW}$. Justify, without calculating the vertex of $P(t)$ again, why the time of maximum solar production of the panels coincides exactly with the time of maximum surplus feed-in.
3. c) Calculate and interpret the average rate of change of the surpluses on the interval $[10, 13]$ and on the interval $[13, 16]$. Based on the results and their signs, explain how the rate of surplus generation changes before and after solar noon.