Exercises: Curvature: Concavity and convexity. Point of inflection

Given the polynomial function $f(x) = x^3 - 6x^2 + 9x + 1$, carry out the following steps to study its curvature:

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

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

3. Find the coordinates of the point of inflection.

Observe the following graph of a function $f(x)$ and visually identify in which intervals the function is convex (upward curvature, $\cup$) and in which it is concave (downward curvature, $\cap$). Indicate approximately the x-coordinate $x$ where you think the point of inflection is located.

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 feet. The designer needs to know where the ramp changes from being "valley-like" (convex) to "mountain-like" (concave) to ensure the transition. Calculate the point of inflection of this path.

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 it is not a point of inflection? Justify your answer by analyzing 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, indicate 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 the function be $f(x) = \frac{x^2}{x-1}$. Although it is a rational function, its curvature study in the domain of definition is simple.

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

2. Study its sign in the corresponding intervals (bearing in mind that $x=1$ does not belong to the domain).

3. Does it have points of inflection? Justify whether the change of curvature at the vertical asymptote is considered a point of inflection.

Given the rational function $f(x) = \frac{x^2-1}{x^2+3}$, carry out a detailed study of its curvature. To do this:

1. Find the intervals of concavity and convexity.

2. Determine the exact coordinates of its points of inflection, if they 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 slope 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 value of the x-coordinate $x$ where the ramp changes from being concave to being convex (point of inflection). Justify why at that 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 approximately follows 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 $t$) when the user growth begins to slow down (point of inflection).

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

Let the piecewise function be:

Calculate the values of $a$ and $b$ so that the function is differentiable at $x = 0$. Once calculated, check if there is a point of inflection at the origin.

Given the function $f(x) = x \cdot e^{-x^2}$, find:

1. The intervals of concavity and convexity.

2. The points of inflection of the curve.

3. What happens to the curvature of the function as $x \to \infty$? Reason 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 more points of inflection in its domain.

Consider the family of functions of the form $f(x) = (x^2 + ax + b) \cdot e^{-x}$, 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 an inflection point 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 and convexity of the function and check if there are any additional inflection points.

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

1. a) Calculate the value of $k$ so that the ramp changes its curvature (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 convex ($\cup$) throughout its domain of definition? Justify your answer analytically.

Given the function $f(x) = \frac{x^3}{x^2 + a}$, where $a$ is a nonzero 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 curvature 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 over all $\mathbb{R}$. From the visual information provided, and restricting yourself to the plotted interval $-2 < x < 2$:

1. a) Determine the intervals of concavity and convexity of the original function $f(x)$.
2. b) Indicate the x-coordinates of the points of inflection of $f(x)$.
3. c) If we know that $f(0) = 0$, sketch the graph of the original function $f(x)$ based on its curvature.

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

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

We want to smooth the junction of two road sections. The first section is given by the function $g(x) = x^2$ for $x \le 1$, and the second section by a third-degree 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 match ($f$ is continuous).

2. The slopes match ($f$ is differentiable).

3. The curvature is the same (the second derivatives match).

Knowing 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 increasing and decreasing intervals, and the relative extrema.
3. c) Analyze the curvature and show that there is a single point of inflection. Determine its exact coordinates.
4. d) What value does the slope of the curve tend to as $x \to \infty$? Does this change the curvature at infinity?

Let the function $f(x) = x^4 \cdot \ln(x)$ be 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, this 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 there is a point of inflection at the limit?