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?

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 behavior of the function on that interval?

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.

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?

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 $\mathbb{R}$.

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 $f(x_1) < f(x_2)$.

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.

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.

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?

A stone is thrown vertically upward. The height in meters 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.

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 pesos, is given by the function $B(x) = -40x^2 + 1\,600x - 6\,000$, where $x$ is the number of batches produced. How many batches should the company produce to maximize its profit? How much is this maximum profit?

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.

A farmer has $120$ meters 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 maximize its area and calculate this area.

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.

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.

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 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 $p(x) = 4\,000 - 10x$ (in pesos), where $x$ is the quantity of units sold. The total production costs follow the function $C(x) = 1\,000x + 40\,000$. Find the profit function $B(x)$, determine the production level that maximizes 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.

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.