Exercises: Increasing and decreasing functions, maxima and minima

From the following graph of the function $f(x)$, 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 the exact coordinates of the points marked A, B, and C. State for each of them whether it is a relative maximum or minimum. Is there any extremum that is also absolute in the represented domain?

Calculate the Average Rate of Change (ARC) of the linear function $f(x) = 4x - 7$ in the interval $[-2, 3]$. From the result obtained, what can you state about the growth of the function in this interval?

The following table of values shows the altitude $h$ (in feet) of a drone in terms of the time $t$ (in seconds) since it starts its flight.

Calculate the ARC in the intervals $[0, 2]$ and $[4, 6]$. Based on these results, is the altitude function increasing or decreasing in these sections?

Given the quadratic function $f(x) = -3x^2 + 12x - 5$, state, without graphing or finding its coordinates, whether the vertex of the parabola corresponds to a relative maximum or minimum. Justify your answer by looking at the sign 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 in those two sections.

Draw in your notebook a continuous graph whose domain is all $\mathbb{R}$, which is strictly increasing in the intervals $(-\infty, -2)$ and $(3, +\infty)$, and strictly decreasing in the interval $(-2, 3)$. According to your drawing, what type of relative extrema (maximum or minimum) will there be at the x-coordinates $x = -2$ and $x = 3$?

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

The height $h$ (in feet) reached by a model rocket in terms of the time $t$ (in seconds) since its launch is given by the expression $h(t) = -5t^2 + 40t + 2$. Calculate at what instant of time the rocket reaches its maximum height and what the exact value of this height is.

The following graph shows the monthly profits $B(t)$ (in thousands of dollars) of a technology company during its first year of existence ($t$ in months, from $t=0$ to $t=12$). Identify the intervals in which the company's profits increase and decrease. In which months are the relative maximum and minimum profits reached?

Given the inverse proportion function $f(x) = \frac{12}{x}$, calculate the Average Rate of Change in the intervals $[1, 3]$ and $[3, 6]$. Prove that, although the numerical values of the ARC are different in each interval, both results justify the decreasing behavior of the function in the interval $(0, +\infty)$.

Consider the piecewise function $f(x)$ defined 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 relative or absolute minimum in its entire domain.

The number of individuals $N$ of an insect population in a laboratory experiment varies in terms of the 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 value and how many individuals make up the population at that exact moment.

The attached graph represents the speed $v$ (in mph) of a vehicle along a journey in terms of the time $t$ (in minutes). State the time intervals in which the vehicle accelerates (increasing speed), brakes (decreasing speed), or maintains a constant speed. What is the maximum speed reached on the journey and for how long is it maintained?

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

• 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 which they are at different times $t$ of their mountain route:

Calculate the Average Rate of Change in the intervals $[0, 2]$, $[2, 4]$, and $[4, 5]$. From the signs of these results, deduce at what approximate time the hiker reached the top of the first mountain they climbed and when they finished descending to the valley before climbing again.

There are $60$ feet of metal fencing available to enclose a rectangular vegetable patch using the straight wall of a barn, so it is only necessary to fence three of its sides. If we call $x$ the length of the two sides perpendicular to the barn wall, find the quadratic function $A(x)$ that expresses the area of the vegetable patch in terms of $x$. What must the exact dimensions of the vegetable patch be for its area to be the maximum possible?

The profit $B(x)$, in dollars, obtained by a factory from the sale of $x$ units of an exclusive product is given by the quadratic function with parameters $B(x) = -ax^2 + bx - 500$. It is known through a market study that the maximum profit is reached when exactly $40$ units are sold 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 polynomial function of degree 4, $f(x)$, which has a "W" shape. From observing the graph, estimate the coordinates of its relative extrema (maxima and minima). Then, knowing that the function is perfectly symmetric 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 $-1$. Once the value of $k$ has been found, justify algebraically whether the function is increasing or decreasing in the entire interval $(-2, +\infty)$.

The energy consumption $C(v)$ (in kWh per 100 miles) of a new electric vehicle in terms of its constant speed $v$ (in mph) can be approximately modeled by the quadratic expression $C(v) = 0.02v^2 - 1.6v + 45$, valid for speeds in the range of $20$ to $100$ mph. At what exact speed must the vehicle travel for the energy consumption to be the minimum possible? What is that minimum consumption?

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

Two rescue boats leave from the same point on the high seas at the same time. Boat A travels in a straight line toward the north at a constant speed of $20$ mph, and boat B travels toward the east at a constant speed of $15$ mph. The squared distance between the two boats in terms of the 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 generic 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 represents 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 $f(x)$ increases; if it is below, it decreases. 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 relative extremum? Justify whether it is a maximum or a minimum.