Curvature and complete graph sketching of functions

Given the function $f(x) = x^3 - 6x^2$, use the second derivative to determine:

1. The intervals of concavity and convexity.

2. The coordinates of the point of inflection.

True or false? Reason your answer: "If $f''(2) = 0$, then $f$ has a point of inflection at $x = 2$." Illustrate your answer with the function $f(x) = x^4$ at $x = 0$.

The temperature of a chemical reaction has been monitored during the first 6 minutes. The function describing the temperature (in $^\circ C$) is $T(t) = -t^3 + 9t^2 - 15t + 40$.

Carry out a study of the function over the interval $t \in [0, 6]$ indicating:

1. At which moments does the temperature increase and at which does it decrease?

2. Find the maximum and minimum temperature reached in that interval.

3. Determine the moment at which the rate of change of the temperature changes trend (point of inflection).

4. Draw the graph.

You can do it on paper and send a photo with your answer.

The temperature of a liquid in an experiment follows, in degrees Celsius, the function $T(t) = t^3 - 6t^2 + 9t + 1$, where $t$ is in hours and $0 \leq t \leq 4$.

• a) Study in which intervals the temperature rises and in which it falls.

• b) Find the point of inflection and interpret what happens to the rate of change of temperature at that moment.

A market study indicates that the monthly profit $B(x)$ (in thousands of euros) of a mobile app depends on advertising expenditure $x$ (in thousands of euros) according to the function:

$B(x) = \ln(x^2 - 2x + 5) - \dfrac{x}{4}$ for $x \in [0, 10]$.

Study the viability of the advertising expenditure by following these steps:

1. a) Justify the domain of the function in the context of the problem.
2. b) Find the advertising expenditure that maximises the profit.
3. c) Investigate whether there are intervals of "diminishing returns", i.e. where the function is concave ($\cap$).
4. d) Draw the graph of the profit function over the given interval.

You can do it on paper and send a photo with your answer.

Carry out a complete study of the function $f(x) = \frac{x^2}{x - 1}$ and sketch it: domain, axes intercepts, asymptotes (vertical and oblique), intervals of monotonicity, relative extrema, curvature and graph.

A biotechnology company has modelled the growth of a bacterial population in a controlled environment using the function $P(t) = 10 \cdot t^2 \cdot e^{-0,5t}$, where $P$ is the number of bacteria in thousands and $t$ is the time in days since the start of the experiment ($t \geq 0$).

Carry out a comprehensive study of the function including:

1. Axes intercepts.

2. Long-term behaviour (horizontal asymptotes).

3. Intervals of increase and decrease, determining when the population is at its maximum.

4. Intervals of concavity and convexity, locating the points of inflection where the rate of growth begins to decrease.

5. A graphical sketch of the population's evolution.

Given the function $f(x) = \dfrac{x^3}{x^2 - k}$, answer the following:

1. a) Find the value of $k$ such that the function has an inflection point at $x = \sqrt{3}$.
2. b) For $k = 1$, carry out the complete analysis of the function: domain, symmetry, asymptotes (vertical and oblique), relative extrema, and curvature.
3. c) Sketch the graph detailing the position of the curve relative to its asymptotes.

You can do it on paper and send a photo with your answer.

Let the piecewise function be defined as:

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

1. a) Study the continuity at $x = 0$.

1. b) For the first piece ($x \le 0$), find the vertex of the parabola and the intercepts with the axes.

1. c) For the second piece ($x > 0$), find its asymptotes.

1. d) Graph the function.

You can do it on paper and send a photo with your answer.

Consider the modulus function $f(x) = \dfrac{|x^2 - 1|}{x}$.

1. Express the function as a piecewise function.

2. Study the continuity and differentiability of the function across its entire domain.

3. Determine the asymptotes and the position of the curve relative to them.

4. Find the intervals of monotonicity and the relative extrema.

5. Draw an accurate graph, paying special attention to the points where the function changes its definition.

The figure shows the graph of a function with its point of inflection indicated.

• a) State the intervals of concavity and convexity of the function.

• b) Sketch the sign of the second derivative in each interval.

• c) Explain what happens to the gradient of the curve as it passes through the point of inflection.

You can do it on paper and send a photo with your answer.

The figure shows the complete graph of a function.

From the graph, carry out a complete study of it:

• a) Domain and axes intercepts.

• b) Asymptotes and infinite branches.

• c) Intervals of monotonicity and relative extrema.

• d) Intervals of concavity and convexity and points of inflection.

• e) Propose a possible analytical expression for the function and justify why it fits all the observed features.