Concavity and Inflection Points

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.

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.

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.

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.

A skatepark designer is modeling 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 meters. The designer needs to know where the ramp changes from "mountain-shaped" (concave) to "valley-shaped" (convex) to ensure a smooth transition. Calculate the inflection point of this trajectory.

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.

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".

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.

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 $x = 0$.

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.

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.

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.

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.

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 $d$.

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.

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?

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.

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 behavior of the curvature as $x$ approaches $0$ from the right. Can we say that there is a point of inflection at the limit?

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 $f(x)$.