Exercises: Monotonicity and extrema of a function

Calculate the Average Rate of Change (ARC) of the function $f(x) = x^2 - 4x + 3$ in the interval $[2, 5]$. From the result, can you deduce whether the function is, on average, increasing or decreasing in that interval?

Observe the following graph of a polynomial function. Identify the coordinates of the points $A$ and $B$, and classify them as relative maxima or 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 minimum of the function.

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 over its entire domain $\mathbb{R}$.

The following graph represents a quadratic function. Determine the coordinates of its vertex $V$. Is it a maximum or a minimum? Is it relative, absolute, or both?

Calculate the Average Rate of Change of the rational function $f(x) = \frac{4}{x}$ in the interval $[1, 4]$. What does the sign of the result indicate about the behavior of the function in that interval?

A stone is thrown vertically upwards. The height in feet 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.

Check if 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 example $x = 2.9$ and $x = 3.1$.

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.

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 if there is any maximum. Each grid square represents $1$ unit on both axes.

Find the value of the parameter $k$ so that the quadratic function $f(x) = -x^2 + kx + 5$ reaches its maximum value exactly 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 dollars, is given by the function $B(x) = -2x^2 + 80x - 300$, where $x$ is the number of batches produced. How many batches must the company produce to maximize its profit? What does this maximum profit amount to?

Prove algebraically that the function $f(x) = 2x^3 + 5$ is strictly increasing for all $x \in \mathbb{R}$. To do this, take any two values $x_1$ and $x_2$ such that $x_1 < x_2$ and check that $f(x_1) < f(x_2)$.

The temperature inside a chemical reactor during the first few hours of a process follows the function $T(t) = -0.2t^2 + 2.4t + 20$, with $t$ in hours and $T$ in degrees Fahrenheit. Determine the maximum temperature reached and the time interval in which the temperature of the reactor is decreasing.

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 a greater average growth in that interval?

Determine the intervals of monotonicity (increasing and decreasing) of the rational function $f(x) = \frac{-3}{x-2}$. You can justify your answer by analyzing the Average Rate of Change in generic intervals or by relying on the graphical properties of hyperbolas.

A farmer has $120$ feet of fencing to enclose a rectangular vegetable patch. One of the sides of the patch will be against the straight wall of a barn, so no fencing will be needed on that side. Find the dimensions of the patch that make its area maximum and calculate this area.

It is known that a quadratic function of the form $f(x) = ax^2 + bx + c$ reaches a relative maximum at the point $(3, 10)$ and intersects 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.

Prove 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 $f(x) \le f(0)$ is satisfied for any real number $x$.

A factory sells a certain product at a unit price that depends on demand, given by the function $p(x) = 200 - 0.5x$ (in dollars), where $x$ is the quantity of units sold. The total production costs follow the function $C(x) = 50x + 2000$. Find the profit function $B(x)$, determine the production level that maximizes the profit and calculate the maximum profit.

The following graph represents a rational function with a vertical and an oblique asymptote. Observing the graph, identify the approximate coordinates of its relative extrema (local maximum and minimum). Then, write the intervals in which the function is increasing and in which it is decreasing.

Given the piecewise function:

$f(x) = \begin{cases} -x^2 + 6x - 5 & \text{if } x \le 4 \\ 2x - 5 & \text{if } x > 4 \end{cases}$

Analyze analytically its intervals of monotonicity and find the coordinates of its relative or absolute extrema. Justify whether the point where the branch changes ($x=4$) represents any type of extremum.

The concentration of a medicine in the blood, measured in mg/L, as a function of the hours elapsed since its administration, is modeled 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 analytically the change in monotonicity.

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.