Exercises: Growth, maximums and minimums

From the graph of the function $f(x)$ below, determine the intervals where the function is strictly increasing and where it is strictly decreasing.

Look at the graph of the function $g(x)$ and write down the exact coordinates of the points labelled A, B and C. State whether each point is a local maximum or local minimum. Are any of these turning points also absolute extrema on the domain shown?

Calculate the average rate of change of the linear function $f(x) = 4x - 7$ in the interval $[-2, 3]$. Based on your result, what can you deduce about whether the function is increasing or decreasing in this interval?

The following table of values shows the altitude $h$ (in metres) of a weather balloon as a function of time $t$ (in seconds) since launch.

Calculate the average rate of change in the intervals $[0, 2]$ and $[4, 6]$. Based on these results, is the altitude function increasing or decreasing over these intervals?

Given the quadratic function $f(x) = -3x^2 + 12x - 5$, state, without plotting the graph or calculating its coordinates, whether the vertex of the parabola corresponds to a local maximum or a local minimum. Justify your answer by looking at the signs of the coefficients of the function.

Consider the function $f(x) = x^2 - 9$. Calculate the average rate of change in the interval $[-4, -1]$ and in the interval $[1, 4]$. Relate the sign of each result to the behaviour (increasing or decreasing) of the function over these two intervals.

Sketch a continuous graph in your notebook whose domain is all of $\mathbb{R}$, which is strictly increasing in the intervals $(-\infty, -2)$ and $(3, +\infty)$, and strictly decreasing in the interval $(-2, 3)$. Based on your sketch, what type of local extrema (maximum or minimum) will occur at the x-coordinates $x = -2$ and $x = 3$?

You can do it on paper and send a photo with your answer.

The temperature of an industrial oven starts at $20 \ ^\circ\text{C}$. During the first $15$ minutes it rises uniformly until reaching $220 \ ^\circ\text{C}$. It remains at this constant temperature for $45$ minutes to bake some parts, and is then switched off, with the temperature falling uniformly to $20 \ ^\circ\text{C}$ over $30$ minutes. State the time intervals over which the temperature is increasing, decreasing, and constant, as well as the absolute maximum value reached by the function.

Seán is analyzing the quadratic function $f(x) = x^2 - 6x + 5$, which describes the height in meters of a skater on a concave track depending on their horizontal position $x$. After performing his calculations, Seán finds the coordinates of the vertex, which are $(3, -4)$, and writes the following conclusion in his notebook: "Since the parabola opens upwards, the function has a relative minimum of value $x = 3$. Therefore, the function is decreasing on the interval $(-\infty, -4)$ and increasing on $(-4, +\infty)$". Identify the two conceptual errors made by Seán when interpreting the coordinates of the vertex, explain in detail why these common confusions occur, and write the correct answer indicating the value of the minimum and the intervals of increase and decrease.

According to records from a Met Éireann weather station in the mountains, the temperature $T$ (in degrees Celsius) during a morning is given by the model $T(t) = -0.2t^2 + 2.4t + 3$, where $t$ is the time in hours elapsed from 6:00 ($t = 0$) to 12:00 ($t = 6$). a) Without calculating specific values yet, look at the shape of the parabola and its vertex: do you expect the average rate of change $\text{TVM}[0, 2]$ (from 6:00 to 8:00) to be greater than, equal to, or less than $\text{TVM}[4, 6]$ (from 10:00 to 12:00)? Justify your physical and graphical prediction qualitatively. b) Calculate both average rates of change accurately, interpreting their units ($^\circ\text{C}/\text{h}$), and check whether your initial estimate was correct.

The height $h$ (in metres) reached by a fireworks rocket as a function of time $t$ (in seconds) since launch is given by the expression $h(t) = -5t^2 + 40t + 2$. Calculate the exact time at which the rocket reaches its maximum height and the exact value of this height.

The following graph shows the monthly subscribers $S(t)$ (in thousands) of a digital magazine during its first year of operation ($t$ in months, from $t=0$ to $t=12$). Identify the intervals over which the number of subscribers increases and decreases. In which months are local maximums and minimums reached?

Given the inverse proportion function $f(x) = \frac{12}{x}$, calculate the average rate of change in the intervals $[1, 3]$ and $[3, 6]$. Verify that, although the numerical values of the average rate of change are different in each interval, both results confirm the decreasing behaviour of the function in the interval $(0, +\infty)$.

Consider the function $f(x)$ defined piecewise as follows:

$f(x) = \begin{cases} -2x + 6 & \text{if } x < 2 \\ x^2 - 8x + 14 & \text{if } x \ge 2 \end{cases}$

Find the coordinates of the vertex of the parabolic section and determine whether the function has any local or absolute minimum over its entire domain.

The number of individuals $N$ in an insect population in a laboratory experiment varies with the number of days $t$ elapsed according to the quadratic model $N(t) = 3t^2 - 24t + 150$, valid for $t \ge 0$. Determine on which day the insect population reaches its minimum size and how many individuals make up the population at that exact moment.

The accompanying graph represents the speed $v$ (in km/h) of a vehicle along a journey as a function of time $t$ (in minutes). State the time intervals over which the vehicle accelerates (increasing speed), decelerates (decreasing speed), or maintains a constant speed. What is the maximum speed reached during the journey and for how long is it maintained?

Sketch a continuous function $f(x)$ in your notebook that satisfies all of the following conditions simultaneously:

• Its domain is the interval $[-5, 6]$.

• Its range is the interval $[-2, 5]$.

• It is strictly increasing in $[-5, -1] \cup [3, 6]$ and strictly decreasing in $[-1, 3]$.

• It passes through the points $(-1, 5)$ and $(3, -2)$.

What do the points $(-1, 5)$ and $(3, -2)$ represent mathematically on the graph of this function?

A hiker records the altitude $h$ at different times $t$ during a mountain walk:

Calculate the average rate of change in the intervals $[0, 2]$, $[2, 4]$ and $[4, 5]$. From the signs of these results, deduce approximately when the hiker reached the summit of the first mountain climbed and when they finished descending into the valley before climbing again.

Determine with justification the algebraic expression of a quadratic polynomial function $f(x) = ax^2 + bx + c$ that simultaneously satisfies three conditions: 1) it has an absolute extremum at the point $V(2, 5)$; 2) it is strictly increasing on the interval $(-\infty, 2)$ and strictly decreasing on $(2, +\infty)$; 3) it intersects the y-axis at the point $(0, 1)$. Justify how you deduce the sign of the leading coefficient $a$, how you use the vertex formula or the vertex form of the parabola to set up the necessary equations, and find the exact values of the coefficients $a$, $b$, and $c$.

The fuel consumption $C(v)$ (in liters per $100\text{ km}$) of a passenger car driving on a highway at a constant speed $v$ (in $\text{km}/\text{h}$, for $60 \le v \le 140$) is modeled by the quadratic function $C(v) = 0.001v^2 - 0.18v + 12.6$. a) Model the most economical driving: calculate at what cruising speed fuel consumption is minimal and what the exact value of said minimum consumption is. b) From the intervals of increase and decrease, justify why driving at $120\text{ km}/\text{h}$ consumes more than driving at $90\text{ km}/\text{h}$. Calculate the average rate of change $\text{TVM}[90, 120]$ and interpret its practical meaning by explaining how many extra liters the car consumes for each $\text{km}/\text{h}$ that exceeds the optimal speed in that section.

A total of $60$ metres of metal fencing is available to enclose a rectangular vegetable garden against the straight wall of a barn, so only three sides need fencing. If $x$ is the length of the two sides perpendicular to the barn wall, find the quadratic function $A(x)$ that expresses the area of the garden in terms of $x$. What must be the exact dimensions of the garden for its area to be as large as possible?

The profit $B(x)$, in euros, earned by a factory from selling $x$ units of an exclusive product is given by the quadratic function with parameters $B(x) = -ax^2 + bx - 500$. A market study shows that the maximum profit is achieved when selling exactly $40$ units and that this maximum profit is $1100$ €. Set up a system of equations and calculate the values of the parameters $a$ and $b$.

The following graph corresponds to a degree 4 polynomial function, $f(x)$, which has a W-shape. By observing the graph, estimate the coordinates of its local extrema (maximums and minimums). Next, knowing that the function is perfectly symmetrical with respect to the y-axis (it is an even function), reason and prove whether the average rate of change in the symmetric interval $[-3, 3]$ will be positive, negative, or zero.

Given the rational function with an unknown parameter $f(x) = \frac{k}{x+2}$, find the exact value of $k$ knowing that the average rate of change of the function in the interval $[0, 2]$ is equal to $-1$. Having found the value of $k$, justify algebraically whether the function is increasing or decreasing across the entire interval $(-2, +\infty)$.

The energy consumption $C(v)$ (in kWh per 100 km) of a new electric vehicle as a function of its constant speed $v$ (in km/h) can be approximately modelled by the quadratic expression $C(v) = 0.02v^2 - 1.6v + 45$, valid for speeds in the range from $20$ to $100$ km/h. At what exact speed should the vehicle travel for its energy consumption to be as low as possible? What is this minimum consumption?

Consider the function defined using a modulus $f(x) = |x^2 - 6x + 5|$. Knowing that the inner parabola $y = x^2 - 6x + 5$ crosses the x-axis at $x=1$ and $x=5$, express $f(x)$ as a piecewise function without using the modulus symbol. Then, find the coordinates of the three local extrema of $f(x)$ and state its intervals of monotonicity.

Two rescue boats set off from the same point in the open sea at the same time. Boat A travels in a straight line North at a constant speed of $20$ km/h, and boat B travels East at a constant speed of $15$ km/h. The squared distance between both boats as a function of time $t$ (in hours) is given by the function $D(t) = (20t)^2 + (15t)^2$. Prove algebraically that the actual distance between them is a strictly increasing function for any time $t > 0$ by calculating the general Average Rate of Change in an interval $[t_1, t_2]$ (with $0 < t_1 < t_2$).

The attached graph does not represent a normal function $f(x)$, but rather the value of its Instantaneous Rate of Change (or rate of growth) at each point $x$. That is, if the graph shown is above the x-axis, the original function $f(x)$ is increasing; if it is below, it is decreasing. From this graph, deduce in which intervals the original function $f(x)$ is increasing and decreasing. At what exact value of $x$ will $f(x)$ reach a local extremum? Justify whether it is a maximum or a minimum.

Consider the piecewise-defined function: $f(x) = 3 - x^2$ if $x \le 1$, and $f(x) = 4x - 2$ if $x > 1$. Two students discuss whether there is a relative extremum at the transition point $x = 1$: Seán calculates the vertices of each piece separately; since the first piece has its vertex at $x = 0$ and the second is a straight line without a vertex, he states that there cannot be any extremum at $x = 1$. Niamh proposes checking first whether the function is continuous at $x = 1$ and then analyzing the sign of the growth immediately to the left and to the right of $x = 1$. a) Which of the two methods is mathematically rigorous? Explain precisely the flaw in Seán's method based on the definition of a relative extremum in a neighborhood. b) Apply the valid method to classify with justification what happens at $x = 1$ (relative maximum, relative minimum, or neither).

Design the complete analytic expression of a piecewise function $g(x)$, continuous on all of $\mathbb{R}$, formed exclusively by two lines (linear functions) of the form: $g(x) = m_1 x + n_1$ if $x < 3$, and $g(x) = m_2 x + n_2$ if $x \ge 3$, satisfying all three of the following conditions simultaneously: 1) At $x = 3$ the function reaches a relative maximum of value $g(3) = 4$. 2) The average rate of change on the interval $[-1, 3]$ is exactly $2$. 3) To the right of $x = 3$, the function decreases twice as fast as it increases to its left. Calculate the coefficients $m_1$, $n_1$, $m_2$, and $n_2$ with justification. Finally, explain why a relative maximum appears at $x = 3$ even though no linear function possesses relative extrema on its own.