Concavity and Inflection Points

A 3D printing company has fixed monthly costs of $\$1000$ and a production cost of $\$5$ for each part manufactured. The total cost function is given by $C(x) = 5x + 1000$.

Does this function have concavity, convexity, or inflection points? Justify your answer based on the graphical representation of the function.

Classify the following quadratic functions as concave ($\cap$) or convex ($\cup$) without graphing them. Remember to look at the leading coefficient.

1. a) $f(x) = 3x^2 - 2x + 1$
2. b) $g(x) = -x^2 + 5x$
3. c) $h(x) = \frac{1}{2}x^2 - 4$
4. d) $i(x) = 8 - 2x^2$

During a basketball game, a player throws the ball toward the basket. The trajectory of the ball follows the equation $h(t) = -5t^2 + 6t + 2$, where $h$ is the height in feet and $t$ is the time in seconds.

Is the trajectory described by the ball concave ($\cap$) or convex ($\cup$)? Is there any inflection point in the flight of the ball? Reason your answer based on the shape of the graph.

A skating ramp (half-pipe type) has a cross section whose shape can be modeled mathematically by the parabola $y = 0.5x^2$.

What type of curvature does the ramp present, concave ($\cap$) or convex ($\cup$)? Is there any inflection point in this continuous structure?

Determine the values of the parameter $k$ for which the quadratic function $f(x) = (2k - 8)x^2 + 3x - 5$ is a concave parabola ($\cap$).

For what exact value of $k$ would the function cease to have curvature? What type of function would be obtained in that specific case?

Prove whether the sum of two convex functions ($\cup$) always results in another convex function.

To investigate this algebraically, take the parabolas $f(x) = x^2$ and $g(x) = 2x^2 - 4x + 1$, both convex. Find the expression for the sum function $h(x) = f(x) + g(x)$ and analyze the type of curvature that $h(x)$ presents.

Consider the family of parametric quadratic functions given by the expression $f(x) = (a^2 - 9)x^2 + 5x - 2$, where $a$ is a real number.

1. a) For what exact values of the parameter $a$ is the function a convex parabola ($\cup$)?
2. b) For what values of $a$ is it a concave parabola ($\cap$)?
3. c) Is there any value of $a$ for which the graph of the function lacks curvature? In that case, what type of geometric function is obtained?

Look at the following graph of a polynomial function of degree 3.

State in which interval the function is concave ($\cap$) and in which it is convex ($\cup$). Identify the exact coordinates of the point $P$, knowing that it is the inflection point and is located at the origin.

State whether the following statement is true or false and justify your answer in detail:

"An inflection point is that point on the graph where the function changes from being increasing to being decreasing, or vice versa."

Given the polynomial function $f(x) = x^3 - 6x^2 + 9x$, it is known from its graph that its inflection point is located exactly at the coordinate $x$ which is the midpoint between its two relative extrema (a local maximum at $x=1$ and a local minimum at $x=3$).

Calculate the exact coordinates of the inflection point and determine the intervals of concavity ($\cap$) and convexity ($\cup$) of the function.

Given the inverse proportion function $f(x) = \frac{3}{x}$, analyze its graph mentally or by drawing a quick sketch.

What type of curvature does it have in the interval $(0, +\infty)$? And in the interval $(-\infty, 0)$? Can it be said that there is an inflection point at $x=0$?

An initial capital of $\$1000$ is invested in a fund at an annual compound interest of $5\%$. The function that describes the accumulated capital over time $t$ (in years) is $C(t) = 1000 \cdot (1.05)^t$.

Knowing that exponential functions with a base greater than $1$ have a characteristic curvature, determine whether this function is concave ($\cap$) or convex ($\cup$). Will there come a time in the future when the rate of capital growth begins to decrease, thus generating an inflection point? Justify your answer.

The following graph corresponds to a piecewise function, composed of two fragments of different parabolas that join smoothly.

Determine the intervals of concavity and convexity of the global function. What does the orange junction point at $x=0$ represent in relation to the curvature of the function?

The function $f(x) = |x^2 - 4|$ represents the absolute value of a parabola and its graph has a rounded "W" shape.

Look at its graph and determine in which intervals the function is concave ($\cap$) and in which it is convex ($\cup$). Can the points $x=-2$ and $x=2$ (marked in red) be considered proper inflection points? Reason your answer.

Given the following piecewise quadratic function:

$f(x) = \begin{cases} -x^2 + 4x & \text{if } x < 2 \\ x^2 - 4x + 8 & \text{if } x \ge 2 \end{cases}$

1. a) Prove algebraically whether the function is continuous at the breakpoint $x = 2$.
2. b) Analytically study the curvature of each of the two sections.
3. c) Can it be rigorously stated that there is an inflection point at $x = 2$? Reason your answer based on the definition of an inflection point.

The following graph represents the altitude of a surveillance drone over time during an ascent maneuver.

Identify what type of curvature the graph has before and after the point $I$. Explain what this inflection point means physically in the context of the drone's flight (how its way of ascending changes).

The following graph shows the total accumulated number of people infected by a virus in a population over the days of a month.

1. a) Visually identify the intervals where the infection curve is convex ($\cup$) and where it is concave ($\cap$).
2. b) Locate the inflection point (marked in orange). What crucial meaning does this point have in the evolution of the epidemic with respect to the number of new daily infections?

Suppose that the rate of change (the growth rate or gradient) of a function $f(x)$ over time is given by the line $y = -2x + 4$.

Knowing that the growth rate of $f(x)$ is decreasing (since the orange line representing the rate is strictly decreasing), what can you deduce about the overall curvature of the original function $f(x)$? Will the original function have any inflection point? Explain your answer.

Imagine that you are filling with water, at a constant rate, a glass bottle that has a wide spherical base and a narrow cylindrical neck.

If we graphically represent the height of the water in the bottle in terms of the elapsed time, describe what the curvature of the graph is like in the lower spherical zone (where the bottle widens and then narrows). Is there any inflection point? Where would it be located approximately?

A high-speed train makes a journey in a straight line. During the first $4$ minutes, it accelerates with increasing intensity. Then, from minute $4$ to $8$, it continues to accelerate but at a decreasing rate until it reaches its maximum cruising speed. Finally, from minute $8$ to $12$, it brakes progressively until it stops at the next station.

If we represent the distance traveled in terms of time, what are the intervals of concavity and convexity of this position graph? Identify the inflection points and explain their physical meaning in relation to the train's acceleration.

A water tank has the shape of an inverted cone (the point downwards and the circular base open upwards). Water begins to be poured into it at a constant rate of gallons per minute.

If we construct a graph that relates the height of the water level $h(t)$ in terms of the elapsed time $t$, will the resulting curve be concave ($\cap$) or convex ($\cup$)? Justify your answer by analyzing how the speed at which the water level rises varies as the cone widens.

Describe and draw a sketch in your notebook of a continuous function $f(x)$ defined in the domain $[-5, 5]$ that simultaneously satisfies all the following conditions:

• It has odd symmetry (symmetry with respect to the origin).

• It has a relative maximum at $x = 2$ and a relative minimum at $x = -2$.

• It has exactly three inflection points located at $x = -3$, $x = 0$ and $x = 3$.

• It is convex ($\cup$) in the intervals $[-5, -3) \cup (0, 3)$.

• It is concave ($\cap$) in the intervals $(-3, 0) \cup (3, 5]$.

The attached graph corresponds to the rational function $f(x) = \frac{100x}{x^2+4}$.

From a detailed observation of the curve and the marked orange points, identify how many inflection points the function has and determine approximately the intervals of concavity ($\cap$) and convexity ($\cup$) in its entire domain $\mathbb{R}$.