Exercises: Curvature and full 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.

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.

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 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.

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.

Marcos has to find the inflection points of the function $f(x) = (x - 3)^4 + 2x - 1$. On his exam, he writes the following working:

"I compute the first derivative: $f'(x) = 4(x - 3)^3 + 2$.

I compute the second derivative: $f''(x) = 12(x - 3)^2$.

I set it equal to zero to find the inflection: $12(x - 3)^2 = 0 \implies x = 3$.

Therefore, the graph of the function has an inflection point at $(3, f(3)) = (3, 5)$".

1. Explain with reasons the conceptual mistake made by Marcos. Why is the condition $f''(x_0) = 0$ necessary but not sufficient to ensure the existence of an inflection point?

2. Correct the exercise by rigorously studying the sign of $f''(x)$ on the entire real line and determine the curvature (concavity and convexity intervals) of the function. Is there ultimately any inflection point?

An agricultural cooperative in Murcia models the expected lemon production $Q(x)$ (in tonnes) as a function of spending on organic fertilisers $x$ (in hundreds of euros) using the function:

The technical management wishes to justify that the function exhibits diminishing returns throughout its domain (that is, that each euro invested brings a smaller harvest increase than the previous euro). Two team members propose two different approaches:

• Method A: Calculate the values $Q(0), Q(1), Q(2), \dots, Q(10)$ and verify that the consecutive increments $\Delta Q = Q(x+1) - Q(x)$ keep decreasing.

• Method B: Analytically calculate the second derivative $Q''(x)$ and determine its sign on the interval $(0, 10)$.

• Apply Method B by determining the sign of $Q''(x)$ and state whether the function is concave or convex.

• Compare both methods and justify which of the two is preferable from a mathematical standpoint to demonstrate the property with general validity on a continuous interval.

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

Carry out a complete study of the function by analysing:

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.

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

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.

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.

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.

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.

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 weekly streams of a newly released song follow the model $S(t) = t \cdot e^{-t/2}$ (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 behaviour (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.

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 study by the INE on digitalisation in small rural municipalities models the proportion of households $P(t)$ (as a decimal fraction) that have a fibre-optic connection after $t$ months from the start of the rollout using the function:

1. Estimate: Before taking derivatives:

a) Reason what maximum proportion of coverage the municipality will approach in the long term (as $t \to \infty$).

b) Estimate whether the rate of new monthly sign-ups will be increasingly higher or increasingly lower as time progresses (saturation effect). Based on that intuition, do you expect the function to be concave ($\cap$) or convex ($\cup$)?

1. Check:

a) Calculate the horizontal asymptote of $P(t)$ using the corresponding limit and compare it with your estimate.

b) Obtain the first and second derivatives of $P(t)$. From the sign of $P''(t)$, formally check the curvature of the function and interpret the result in terms of diminishing returns of the rollout.

An e-commerce company based in Zaragoza analyses the relationship between the daily budget allocated to digital advertising $x$ (in hundreds of euros, with $0 \le x \le 8$) and the number of daily sales achieved $V(x)$, according to the function:

1. Determine the intervals where the function is convex (increasing marginal returns, where each additional euro produces an increasingly larger increase in sales) and where it is concave (diminishing marginal returns or saturation).

2. Calculate the exact coordinates of the inflection point of the function and interpret its economic meaning: why is this point called the "point of maximum marginal efficiency" of the investment?

3. A manager proposes: "To achieve maximum advertising profitability, we should invest exactly the amount that produces the absolute maximum of daily sales". Justify by studying $V'(x)$ and $V''(x)$ why this statement confuses maximum volume with the highest efficiency of spending.

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.

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.

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.

Given the function $f(x) = \sqrt{\dfrac{x^2 - 4}{x^2 - 1}}$, carry out a complete study in order to graph it:

1. a) Determine the domain of definition (be careful with the intervals where the quotient is negative).

1. b) Analyze the symmetries of the curve.

1. c) Calculate the horizontal and vertical asymptotes, studying the behavior of the function near them.

1. d) Find the intervals of increase and decrease. Are there any relative maxima or minima?

1. e) Draw the graph of the function from the previous results (also include the points where it crosses the axes).

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.

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.

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.

It is desired to determine the coefficients of a third-degree polynomial function $f(x) = ax^3 + bx^2 + cx + d$ intended to model an economic cycle. The following information about its graphical representation is available:

1. The graph has an inflection point at $I(1, 4)$.

2. The tangent line to the curve at said inflection point is $y = -3x + 7$.

3. The function has a relative extremum at $x = 0$.

1. a) Set up a system of linear equations with the given conditions and determine the values of the parameters $a$, $b$, $c$, and $d$ with reasons.
2. b) With the obtained function, study the concavity and convexity intervals and classify the relative extremum located at $x = 0$ (maximum or minimum) using the second derivative test.
3. c) Reason theoretically whether any third-degree polynomial function $P(x) = \alpha x^3 + \beta x^2 + \gamma x + \delta$ (with $\alpha \ne 0$) necessarily always has a single inflection point on the entire real line, or whether it could have more than one or none.

The charging power $P(t)$ (in kW) accepted by the battery of an electric vehicle at an ultra-fast charging station depends on the elapsed time $t$ (in minutes, for $0 \le t \le 60$) according to the following piecewise-defined function:

1. Study the continuity and differentiability of $P(t)$ at the transition point $t = 20$. Does the graph have a unique tangent line at that instant, or does it form a corner point?

2. Find the expression for the second derivative $P''(t)$ in each of the sections and determine the concavity and convexity of the curve on $(0, 20)$ and on $(20, 60)$.

3. Reason formally whether the point $(20, P(20))$ is an inflection point of the function, analysing whether the theoretical requirements (continuity, existence of a tangent line, and change of curvature) are strictly met, taking into account that the second derivative is not defined at $t = 20$.

4. Interpret the progression of the recharge: on which interval is the power loss accelerating, and on which is the rate of decline damped, tending to stabilise?