Curvature. Concavity, convexity and the 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.

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.

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 $x = 0$.

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".

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.

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 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.