Local Maxima and Minima (Relative Extrema)

An influencer has found that the number of "likes" on their posts follows the function $L(t) = t^3 - 3t^2 + 4$, where $t$ is the time in hours since publication ($0 \le t \le 3$).

Calculate the critical points of the function in that interval by solving the equation $L'(t) = 0$.

Observe the graph of the function $f(x)$ representing the evolution of the number of active users on a new mobile app during the first 8 hours of the morning. Identify the coordinates $(x, y)$ of the relative maxima and minima shown in the diagram.

Given the following graph of the derivative function $f'(x)$ (a straight line), state for which values of $x$ the original function $f(x)$ has a relative maximum or minimum. Justify your answer based on the sign change of the derivative.

Find the relative extrema of the polynomial function $f(x) = \frac{x^3}{3} - 4x + 5$. State the full coordinates $(x, y)$ of each identified extremum.

The following graph corresponds to the derivative function $f'(x)$ of a continuous function $f(x)$. From the visual information, identify with reasons the x-coordinates (abscissae) of the points where the original function $f(x)$ has its relative maxima and minima. Explain what test or criterion you use to differentiate between a maximum and a minimum based on the sign of $f'(x)$.

Let the rational function be $f(x) = \dfrac{x^2 - 1}{x^2 + 1}$.

Study the increase and decrease of the function and locate its relative extrema. Does this function have any point where the derivative is zero but is neither a maximum nor a minimum? Give reasons for your answer.

A content creator has analysed the evolution of views on their videos during the first 10 hours after publication. The function describing the cumulative number of views (in thousands) is given by $V(t) = -t^3 + 12t^2 + 60t$, where $t$ is the time in hours, with $t \in [0, 10]$.

Calculate the moment when the number of views is growing fastest. To do this, find the value of $t$ that maximises the growth rate function $V'(t)$ and determine the relative extrema of the original function $V(t)$ in that interval.

It is known that a function $f(x)$ has a critical point at $x = 3$. If the second derivative is $f''(x) = 2x - 10$, use the value of $f''(3)$ to determine whether there is a relative maximum or a relative minimum at $x = 3$.

Classify the relative extrema of the function $f(x) = x^2 - 6x + 8$ using the second derivative test. To do this:

1. Find $f'(x)$ and solve for $x$ such that $f'(x) = 0$.

2. Find $f''(x)$ and check its sign at the value obtained above.

Calculate the values of the parameters $a$ and $b$ so that the polynomial function $f(x) = ax^3 + bx^2 + 9x + 4$ has a relative maximum at the point $P(1, 8)$. Once the values have been found, check using the second derivative whether it is indeed a maximum or a minimum.

Given the profit function of a small textile company, defined by $B(x) = -x^2 + 10x - 16$, where $x$ is the number of garments manufactured (in hundreds):

1. Calculate the derivative function $B'(x)$.

2. Find the value of $x$ that makes the derivative zero (critical point).

3. Determine whether this point corresponds to a relative maximum or minimum.

The function $C(x) = 2x^2 - 8x + 20$ describes the maintenance costs of a web server as a function of the workload $x$.

Sketch the parabola and find the value of $x$ that minimises the cost.

A courier company has determined that the average cost of delivering a package depends on the distance $x$ (in km) to the distribution center according to the function:

$C(x) = \dfrac{2x^2 + 32}{x}$ for $x > 0$.

Find the distance $x$ at which the delivery cost is a minimum. Justify that the value obtained is indeed a relative minimum using the first or second derivative test.

A logistics study for a food delivery company estimates that the hourly operating cost depends on the average speed $v$ (in km/h) of its delivery drivers according to the function:

$C(v) = 2v^2 + \frac{16000}{v} + 50$ for $v > 0$.

1. a) Calculate the optimal speed that minimises the operating cost.
2. b) Show that there are no relative maxima for this function in its domain.
3. c) If, due to safety regulations, the average speed cannot exceed 15 km/h, what would be the actual minimum cost the company could achieve?

We wish to design a rectangular billboard with a surface area of $18$ $m^2$. The cost of the top and bottom frame is $400$ pesos per linear meter, while the cost of the sides is $300$ pesos per linear meter. Furthermore, for aesthetic reasons, the height of the billboard cannot be less than $2$ meters or greater than $5$ meters.

1. a) Express the total frame cost function in terms of the width $x$.
2. b) Determine the dimensions of the billboard that minimize the total cost within the given constraints.
3. c) What is this minimum cost?

A marketing department estimates that the monthly profits of an advertising campaign (in pesos) follow the function $B(x) = -x^3 + 15x^2 + 600x - 1000$, where $x$ is the number of media channels used ($x \in [1, 30]$).

Find the number of channels that maximizes the monthly profit. Distinguish between the relative extrema found inside the interval and the values that the function takes at the boundaries of the domain to ensure the absolute maximum.

Find the relative extrema of the function $f(x) = \frac{\ln(x^2)}{x}$. Pay special attention to the domain of the function and its symmetry to simplify the study. Justify whether the extrema found are absolute maxima or minima in the interval $(0, +\infty)$.

The effectiveness of a new drug for the study of memory in teenagers is measured on a scale from 0 to 20. After ingestion, the effectiveness $E$ after $t$ hours is given by the exponential function:

$E(t) = 10t \times e^{-0.2t}$ for $t \ge 0$.

Determine the time $t$ at which the drug reaches its maximum effectiveness. State the value of this maximum effectiveness and confirm whether it is a relative extremum.

Consider the piecewise function:

$f(x) = \begin{cases} x^2 + 4x + 3 & \text{if } x \le 0 \\ \dfrac{3}{x+1} & \text{if } x > 0 \end{cases}$

Study the continuity and differentiability at $x = 0$. Based on the previous study and on calculating the derivative in each section, determine whether the function has a relative maximum or minimum at $x = 0$.

A digital marketing agency has determined that the engagement level of a social media campaign follows the function $E(t) = \frac{at}{1 + bt^2}$, where $t$ is the time in days since the launch. It is known that the maximum engagement level is reached exactly at 4 days and that, at that moment, the engagement value is 10 units.

1. a) Determine the values of the parameters $a$ and $b$.
2. b) Justify, using the second derivative test, that at $t = 4$ there is indeed a relative maximum.
3. c) What happens to the engagement in the very long term? Interpret the result.

Consider the profit function of a tech startup defined piecewise for the first two years ($t \in [0, 24]$ months):

$B(t) = \begin{cases} t^2 - 12t & \text{if } 0 \le t < 12 \\ \ln(t - 11) + k & \text{if } 12 \le t \le 24 \end{cases}$

1. a) Calculate the value of $k$ so that the function is continuous at $t = 12$.
2. b) For the value of $k$ found, study whether the function is differentiable at $t = 12$.
3. c) Determine the intervals of increase and decrease, and find the relative extrema (local maxima and minima) of the function over the entire interval.

Given the polynomial function $f(x) = x^3 + ax^2 + bx + c$, find the values of $a$, $b$ and $c$ knowing that the function has a point of inflection at $x = 1$, a relative extremum at $x = 2$, and that the graph passes through the point $(0, 5)$. Once the parameters have been found, classify all the relative extrema of the function.

An "influencer" analyzes the virality of his videos. The number of daily views $V$ (in thousands) after $t$ hours since posting, with $0 \le t \le 24$, follows the model $V(t) = t^2 \times e^{-t/4}$. The figure shows the graph of the function.

1. a) At what exact moment of the day does the video reach its peak number of views?

1. b) Determine the time intervals in which the growth rate of the views is increasing (use the second derivative).

1. c) Calculate the maximum value of daily views.

Let the function be $f(x) = |x^2 - 4| \times e^x$.

1. a) Express the function without modulus signs.
2. b) Study the differentiability at the points $x = -2$ and $x = 2$.
3. c) Determine all relative maxima and minima of the function. State which of them correspond to points where the function is not differentiable (sharp points).