Curvature and Complete Function Sketching

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.

Given the function $f(x) = x^4 - 6x^2$:

• a) Calculate its second derivative.

• b) Determine the intervals of concavity ($\cap$) and convexity ($\cup$).

• c) Find the coordinates of its points 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 figure shows the graph of a cubic polynomial function.

From the graph, and without knowing the formula:

• a) State the intervals of increase and decrease.

• b) Indicate where there is a relative maximum and minimum.

• c) State the intervals of concavity and convexity and where the point of inflection is located.

For each case, sketch a function that satisfies the following:

• a) Increasing and convex ($\cup$) across its entire domain.

• b) Decreasing and concave ($\cap$) across its entire domain.

• c) Increasing up to $x = 1$, decreasing from $x = 1$, and convex across its entire domain.

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

Consider the function $f(x) = x + e^{-x}$.

• a) Calculate $f'$ and $f''$.

• b) Prove that the function is convex ($\cup$) across all of $\mathbb{R}$.

• c) Find its only relative extremum and classify it.

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.

Sketch a continuous function defined on all of $\mathbb{R}$ that is increasing and concave ($\cap$) across its entire domain. Check that the function $f(x) = -e^{-x}$ satisfies these conditions by calculating its first two derivatives.

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

A market study indicates that the monthly profit $B(x)$ (in thousands of pesos) of a mobile app depends on advertising expenditure $x$ (in thousands of pesos) 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 maximizes 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.

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.

Determine a cubic polynomial function $f(x) = ax^3 + bx^2 + cx + d$ that satisfies the following conditions simultaneously:

• It has a point of inflection at $I(0, 1)$.

• The tangent line to the graph at that point of inflection is $y = -3x + 1$.

• It has a relative extremum at $x = 1$.

After finding the expression of the function, carry out a complete study of it (intercepts, extrema, curvature) and represent it graphically.

Determine the coefficients $a$ and $b$ of the function $f(x) = x^3 + ax^2 + bx$ knowing that it has a point of inflection at the point $(1, 2)$.

• a) Set up the conditions that must be satisfied and find $a$ and $b$.

• b) Using the function obtained, carry out a complete study of it: axes intercepts, monotonicity, extrema, curvature, and sketch.

A financial analyst studies the evolution of the expected profit of a tech startup (in thousands of pesos) using the function $f(x) = \frac{80x^2}{x^2+4}$, where $x$ represents time in years since its founding ($x \ge 0$).

Carry out a complete study of the function by analyzing:

1. Domain and axes intercepts.

2. Asymptotes and position of the curve relative to them.

3. Intervals of increase, decrease, and existence of maxima or minima.

4. Curvature (concavity and convexity).

5. Sketch the graph of the function based on the previous results.

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.

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.

Consider the function $f(x) = \ln(x^2 - 4x + 5)$.

1. Determine its domain of definition (reason whether it is all of $\mathbb{R}$).

2. Study whether it exhibits any symmetry.

3. Find the intervals of increase and decrease, and the coordinates of the relative extremum.

4. Study the curvature and the points of inflection.

5. Sketch its graph.

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

The weekly streams of a newly released song follow the model $S(t) = t \times e^{-t/2}$ (in thousands), where $t \geq 0$ is in weeks.

• a) Find the week with the highest number of streams and calculate this maximum.

• b) Study the long-term behavior (horizontal asymptote) and interpret it.

• c) Study the curvature: when does the decline in streams begin to slow down?

• d) Sketch the complete graph.

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

A biotechnology company has modelled the growth of a bacterial population in a controlled environment using the function $P(t) = 10 \times t^2 \times e^{-0.5t}$, 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 behavior (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.

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.

Draw the graph of a function $f(x)$, continuous on $\mathbb{R} \setminus \{2\}$, that simultaneously satisfies all of the following conditions:

• $\displaystyle \lim_{x \to -\infty} f(x) = 0$ and $\displaystyle \lim_{x \to +\infty} f(x) = +\infty$.

• It has a vertical asymptote at $x = 2$, such that $\displaystyle \lim_{x \to 2^-} f(x) = +\infty$ and $\displaystyle \lim_{x \to 2^+} f(x) = -\infty$.

• It passes through the point $(0, 4)$.

• It has a relative minimum at the point $(4, 1)$.

• It is convex ($\cup$) on the interval $(-\infty, 2)$.

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

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.