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 meters) 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 behavior (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.

The height $h$ (in meters) 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 behavior 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.

A total of $60$ meters 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 pesos, earned by a factory from selling $x$ units of an exclusive product is given by the quadratic function with parameters $B(x) = -ax^2 + bx - 10\,000$. A market study shows that the maximum profit is achieved when selling exactly $40$ units and that this maximum profit is $22\,000$ pesos. 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.