Concavity: concave up, concave down and points of inflection

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.

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.

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.

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 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.

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.

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.

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".

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?

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.

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.

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.

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 $f(x)$.

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.