Exercises: Concavity: Concave up, concave down and points of inflection

Given the polynomial function $f(x) = x^3 - 6x^2 + 9x + 1$, carry out the following steps to study its curvature (the region where $f''(x)<0$ is called concave ($\cap$) and the region where $f''(x)>0$ is called convex ($\cup$)):

1. a) Calculate the second derivative $f''(x)$.

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

1. c) Find the coordinates of the inflection point.

Look at the following graph of a function $f(x)$ and visually identify the intervals on which the function is convex (curving upwards, $\cup$) and those on which it is concave (curving downwards, $\cap$). State approximately the x-coordinate $x$ where you think the point of inflection lies.

A skatepark designer is modelling the profile of a ramp using the function $f(x) = \frac{1}{6}x^3 - x^2 + 5$, where $x$ represents the horizontal distance in metres. The designer needs to know where the ramp changes from "mountain-shaped" (concave) to "valley-shaped" (convex) to ensure a smooth transition. Calculate the point of inflection of this trajectory.

Given the function $f(x) = x^4 - 2x^3$, study its curvature. Does this function have any point where the second derivative is zero but is not a point of inflection? Justify your answer by analysing the sign of $f''(x)$ on both sides of the singular points of the second derivative.

Match each of the following conditions on the second derivative with the shape of the graph of the function $f(x)$ in a given interval:

1. $f''(x) > 0$ for all $x \in (a, b)$

2. $f''(x) < 0$ for all $x \in (a, b)$

3. $f''(c) = 0$ and there is a change of sign of $f''(x)$ when passing through $x = c$.

Use the terms: Concave (i.e. concave down, $\cap$), Convex (i.e. concave up, $\cup$) and Point of Inflection.

Find the value of the parameter $k$ so that the function $f(x) = x^3 + kx^2 + 5x - 2$ has a point of inflection at $x = 2$. Once $k$ is found, state whether the function is concave or convex at the point $x = 0$.

Determine the intervals of curvature of the exponential function $f(x) = e^{2x} - 4x$. Does this function have any point of inflection? Reason your answer based on the value of the second derivative.

Let $f(x) = \frac{x^2}{x-1}$ be a function. Although it is a rational function, studying its curvature on its domain is straightforward. The figure shows its graph and its asymptotes.

1. Find $f''(x)$.

2. Study its sign on the corresponding intervals (taking into account that $x=1$ does not belong to the domain) and relate it to the curvature you see on each branch of the graph.

3. Does it have inflection points? Justify whether the change in curvature at the vertical asymptote counts as an inflection point.

Seán has to study the curvature of the polynomial function $f(x) = x^4 - 4x^3 + 6x^2 + 2$. On his exam, he writes the following reasoning:

"First derivative: $f'(x) = 4x^3 - 12x^2 + 12x$.

Second derivative: $f''(x) = 12x^2 - 24x + 12 = 12(x - 1)^2$.

Setting it equal to zero: $12(x - 1)^2 = 0 \implies x = 1$.

Therefore, the graph of the function changes curvature at $x = 1$ and has an inflection point at $(1, 5)$."

1. a) Identify and explain in detail the conceptual error Seán made. Is the condition $f''(x_0) = 0$ sufficient on its own to guarantee the existence of an inflection point?
2. b) Correct the exercise by determining the actual intervals of concavity ($\cap$) and convexity ($\cup$) of $f(x)$, and conclude with justification whether the graph has inflection points or not.

Design the algebraic expression of a cubic function $f(x) = ax^3 + bx^2 + cx + d$ that simultaneously satisfies the following four conditions:

1. Its graph intersects the y-axis at the point $(0, -5)$.

2. It has an inflection point at the coordinates $(2, 3)$.

3. The tangent line to the graph at said inflection point has a slope equal to $-4$.

4. The function is concave ($\cap$) on $(-\infty, 2)$ and convex ($\cup$) on $(2, +\infty)$.

Calculate with reasoning the coefficients $a, b, c$, and $d$, and check explicitly using the sign of $f''(x)$ that the curvature condition 4 holds strictly with the values you found.

Given the rational function $f(x) = \frac{x^2-1}{x^2+3}$, carry out a detailed study of its curvature (consider it concave $\cup$ where $f''(x)>0$ and convex $\cap$ where $f''(x)<0$). To do so:

1. a) Find the intervals of concavity and convexity.

1. b) Determine the exact coordinates of its inflection points, if any exist.

It is known that the polynomial function $f(x) = x^3 + ax^2 + bx + c$ has a point of inflection at the point $P(1, 2)$ and that the gradient of the tangent line at that point is $m = -3$. Calculate the values of the parameters $a, b$ and $c$.

A team of engineers is designing a jump ramp for an extreme sports park. The profile of the central section of the ramp follows the function $f(x) = (x-2)e^x$ for $x \in [-1, 3]$.

Calculate the x-coordinate $x$ where the ramp changes from concave to convex (point of inflection). Justify why at this point the ramp reaches its maximum downward slope in that section.

Given the function $f(x) = \ln(x^2 + 9)$, determine:

1. The intervals where the function is concave ($\cap$) and where it is convex ($\cup$).

2. The coordinates of the points of inflection.

3. The equation of the tangent line to the graph of $f(x)$ at its point of inflection with a positive x-coordinate.

The number of active users on a new social network during the first 6 months follows approximately the function $N(t) = \frac{30t^2}{t^2 + 4}$, where $t$ is the time in months and $N(t)$ is the number of users in thousands.

1. Calculate the exact moment (value of $t$) at which the growth of users begins to decelerate (point of inflection).

2. Explain the meaning of the curvature of the function before and after that moment in terms of "rate of growth".

Consider the piecewise defined function:

Calculate the values of $a$ and $b$ so that the function is differentiable at $x = 0$ and its graph passes through the point $(\pi, \pi)$. Once calculated, check whether there is an inflection point at the origin.

Given the function $f(x) = x \times e^{-x^2}$, you are asked to:

1. a) Determine the intervals of concavity and convexity by studying the sign of $f''(x)$ (consider the function convex, $\cup$, where $f''(x) > 0$ and concave, $\cap$, where $f''(x) < 0$).

1. b) Find the inflection points of the curve.

1. c) What happens to the curvature of the function when $x \to +\infty$ and when $x \to -\infty$? Justify your answer based on the limit of the second derivative.

Determine the values of $a$ and $b$ so that the function $f(x) = \frac{ax+b}{x^2-4}$ has a point of inflection at $x = 0$ and passes through the point $(1, -1)$. Once found, study whether the function has further points of inflection in its domain.

Given the rational function $f(x) = \frac{12}{x^2 + 3}$:

1. a) Estimate before calculating: Observe that the function is symmetric with respect to the y-axis (even), strictly positive on all of $\mathbb{R}$, reaches a relative maximum at $(0, 4)$, and has a horizontal asymptote at $y = 0$ as $x \to \pm\infty$. Based solely on these qualitative data, deduce with reasoning whether the graph must have inflection points, how many you would expect to find at a minimum, in which quadrants they will be located, and what curvature the graph should have near the origin versus regions very far from it ($|x| \to \infty$).
2. b) Check analytically: Calculate $f''(x)$, determine the intervals of concavity ($\cap$) and convexity ($\cup$), and find the exact coordinates of the inflection points. Check whether the analytical results precisely confirm the deductions you proposed in your previous estimation.

To analyze the curvature and classify the potential inflection points of the function $f(x) = x^5 - 5x^4 + 7$, you calculate its first two derivatives and obtain $f'(x) = 5x^4 - 20x^3$ and $f''(x) = 20x^2(x - 3)$. The candidates for inflection points are $x = 0$ and $x = 3$.

Two classmates propose different solving methods:

• Method 1 (Higher-order derivatives): Calculate $f'''(x)$, $f^{(4)}(x)$, etc., and evaluate at each candidate until finding a higher-order derivative that does not vanish.

• Method 2 (Study of the sign of $f''$): Directly analyze the sign of $f''(x) = 20x^2(x - 3)$ on the intervals determined by the zeros of the second derivative.

1. a) Apply Method 1 to classify $x = 0$ and $x = 3$.
2. b) Apply Method 2 to determine the intervals of concavity and convexity and classify both points.
3. c) Critically compare both methods for this case: which of the two is more efficient and why does the presence of the factor $x^2$ make Method 2 especially advantageous compared to Method 1? What conceptual limitation would Method 1 present if the function did not admit higher-order derivatives at the analyzed point?

Consider the family of functions of the form $f(x) = (x^2 + ax + b) \times e^{-x}$, where $a$ and $b$ are real parameters. It is known that the graph of the function passes through the point $(0, 1)$ and that it has a point of inflection at the point with x-coordinate $x = 1$.

1. a) Determine the values of $a$ and $b$.
2. b) For the values found, study the intervals of concavity of the function and check whether any additional points of inflection exist.

A team of engineers is designing a transition ramp for a new skateboard model. The height of the ramp follows the function $f(x) = \ln(x^2 + kx + 9)$ on the interval $x \in [-2, 2]$.

1. a) Calculate the value of $k$ so that the ramp changes its concavity (has a point of inflection) exactly at $x = 0$.
2. b) Is it possible to find a value of $k$ such that the function is always concave up ($\cup$) across its entire domain of definition? Justify your answer analytically.

Given the function $f(x) = \frac{x^3}{x^2 + a}$, where $a$ is a non-zero real constant:

1. a) Show that if $a > 0$, the function always has exactly three collinear points of inflection.
2. b) If $a = -3$, study the concavity of the function detailing its intervals and points of inflection, taking into account the discontinuities of the function.

The following graph corresponds to the first derivative function, $f'(x)$, of a continuous function $f(x)$ defined on all of $\mathbb{R}$. From the visual information provided, and restricting yourself to the plotted interval $-2 < x < 2$:

1. a) Determine the intervals of concavity (concave up / concave down) of the original function $f(x)$.
2. b) State the x-coordinates of the points of inflection of $f(x)$.
3. c) Given that $f(0) = 0$, sketch the graph of the original function $f(x)$ based on its concavity.

Let $f(x) = \text{arctg}(x^2 - 1)$ be the function.

1. a) Find the fully simplified expression for $f''(x)$.
2. b) Determine the points on the curve where the gradient of the tangent line reaches its local maximum or minimum values. Hint: Remember that this occurs where the concavity changes.
3. c) Study the intervals of concavity of $f(x)$.

It is desired to smooth the junction between two sections of road. The first section is given by the function $g(x) = x^2$ for $x \le 1$, and the second section by a cubic polynomial function $h(x) = ax^3 + bx^2 + cx + d$ for $x > 1$.

For the transition to be "perfectly smooth", the engineers require that at the junction point $x = 1$:

1. The functions coincide ($f$ is continuous).

2. The gradients coincide ($f$ is differentiable).

3. The concavity (curvature) is the same (the second derivatives coincide).

Given that the second section must have a point of inflection at $x = 2$, calculate the values of $a, b, c$ and $d$.

Consider the function $f(x) = \frac{e^x}{x^2}$.

1. a) Study the domain and the asymptotes of the function.
2. b) Determine the intervals of increase and decrease, and the relative extrema.
3. c) Analyse the concavity/convexity and prove that there is no inflection point.
4. d) Towards what value does the slope of the curve tend as $x \to \infty$? Does this change the curvature at infinity?

Let $f(x) = x^4 \times \ln(x)$ be the function defined for $x > 0$.

1. a) Calculate $f'(x)$, $f''(x)$ and $f'''(x)$.
2. b) Determine the intervals of concavity and convexity.
3. c) Is there any point on the graph where the tangent line is horizontal and, simultaneously, that point is a point of inflection? Justify your answer.
4. d) Study the behaviour of the curvature as $x$ approaches $0$ from the right. Can we say that there is a point of inflection at the limit?

According to the projections of the Climate Action Plan in Ireland, the cumulative capacity of solar photovoltaic self-consumption (in gigawatts, $\text{GW}$) between the years 2018 ($t = 0$) and 2026 ($t = 8$) is modeled by the logistic function:

$P(t) = \frac{12}{1 + 8\times e^{-0.5t}}$

where $t$ represents the elapsed time in years since the end of 2018.

1. a) Determine with reasoning the limit of $P(t)$ as $t \to +\infty$ and interpret the value obtained in the context of the model (saturation capacity of installable power).
2. b) Analytically calculate the second derivative $P''(t)$ and find the exact instant $t_0$ at which the function has an inflection point. In roughly what year and month did this milestone occur?
3. c) Study the sign of $P''(t)$ before and after $t_0$. Interpret what meaning convexity and concavity have in terms of the annual installation rate ($P'(t)$): during which period was the installation rate accelerating and when did it begin to slow down, even though total capacity continued to increase?
4. d) Prove algebraically that in any general logistic model of the form $P(t) = \frac{K}{1 + b\times e^{-kt}}$ (with $K, b, k > 0$), the inflection point is always reached at the instant when the quantity reaches half of its saturation limit value ($P(t_0) = \frac{K}{2}$).

In the design of the longitudinal profile of an acceleration ramp for an amusement park ride in Dublin, the starting elevation needs to be smoothly connected to the upper platform. The ramp must be represented by a differentiable function $f(x)$ defined on the interval $[0, 4]$ (where $x$ is the horizontal distance in meters and $f(x)$ is the height in meters) that satisfies:

1. It begins at the point $(0, 0)$ with a horizontal slope ($f'(0) = 0$).

2. It ends at the point $(4, 2)$ also with a horizontal slope ($f'(4) = 0$).

3. To avoid sudden jerks for the passengers, the curvature must vary continuously, changing from convex ($\cup$) to concave ($\cap$), with exactly one inflection point inside the interval $(0, 4)$.

1. a) Formally prove why it is impossible for a second-degree polynomial to satisfy these specifications simultaneously.
2. b) Determine the coefficients of the polynomial of lowest possible degree, $f(x) = ax^3 + bx^2 + cx + d$, that satisfies the position and slope conditions at the endpoints.
3. c) For the cubic function obtained, prove that the curvature condition is verified: determine the intervals of convexity and concavity, find the coordinates of the inflection point, and justify why at the x-coordinate of said inflection point the ramp reaches its maximum slope, calculating its exact value.