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 $L'(t) = 0$.

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.

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.

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.

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 $f'(x)$.

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 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?