Concavity and Points of Inflection
Picture a skateboard ramp that rises from 0 ft to 6 ft. One design gets steeper as you roll up it, like a bowl. Another starts steep and flattens out, like a hill. The first derivative alone can't tell them apart, because the slope is positive on both. The second derivative can, and it also tells you exactly where a ramp switches from one shape to the other.
Concave Up and Concave Down
The first derivative $f'$ is the slope of the graph. The second derivative $f''$ is the derivative of the slope, so it tells you how the slope is changing.
-
If the slope is increasing, the graph bends upward like a cup that holds water. This is concave up ($\cup$).
-
If the slope is decreasing, the graph bends downward like a cup that spills. This is concave down ($\cap$).
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…
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:
-
$f''(x) > 0$ for all $x \in (a, b)$
-
$f''(x) < 0$ for all $x \in (a, b)$
-
$f''(c) = 0$ 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.
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…
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.
Points of Inflection
Notice that the tangent line at an inflection point crosses the graph, because the curve is on one side before the point and on the other side after it.
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.…
Given the polynomial function $f(x) = x^3 - 6x^2 + 9x + 1$, carry out the following steps to study its curvature:
-
Calculate the second derivative $f''(x)$.
-
Determine the intervals of concavity ($\cap$) and convexity ($\cup$).
-
Find the coordinates of the point of inflection.
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?…
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.
Inflection Points and the Steepest Slope
Look at the slope instead of the curve. If $f''$ changes from positive to negative at $c$, then $f'$ goes from increasing to decreasing. So $f'$ has a local maximum at $c$. If $f''$ changes from negative to positive, $f'$ has a local minimum there.
The first curve is $N$ and the second is $N'$. Notice that $N'$ peaks exactly where $N$ switches from concave up to concave down.
Concavity also has a plain-English meaning for rates. Concave up means the rate of change is increasing (growth is speeding up). Concave down means the rate of change is decreasing (growth is slowing down), even if the quantity is still growing.
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…
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.
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…
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.
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…
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.
-
Calculate the exact moment (value of $t$) when the user growth begins to slow down (point of inflection).
-
Explain the meaning of the curvature of the function before and after that instant in terms of "growth rate".
Reading Curvature from the Graph of f'
Sometimes you are given the graph of $f'$ and asked about $f$. Since $f'' = (f')'$, concavity of $f$ is the increasing/decreasing behavior of the graph of $f'$.
| On the graph of $f'$ | Meaning for $f$ |
|---|---|
| $f'$ is increasing (rising) | $f'' > 0$, so $f$ is concave up |
| $f'$ is decreasing (falling) | $f'' < 0$, so $f$ is concave down |
| $f'$ has a peak or a valley | $f$ has a point of inflection |
| $f'$ crosses the $x$-axis | $f$ has a possible local max or min |
Do not confuse the two kinds of special points. Where $f'$ crosses the $x$-axis, the slope of $f$ is zero (possible extremum of $f$). Where $f'$ has a peak or valley, the slope is extreme (inflection of $f$).
To sketch $f$ from $f'$, first mark where $f$ rises or falls (sign of $f'$). Then bend each piece up or down according to the direction of $f'$. Finally, place the graph using any given point such as $f(0) = 0$.
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…
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 $-2 < x < 2$:
- a) Determine the intervals of concavity and convexity of the original function $f(x)$.
- b) Indicate the x-coordinates of the points of inflection of $f(x)$.
- c) If we know that $f(0) = 0$, sketch the graph of the original function $f(x)$ based on its curvature.
Exponential and Logarithmic Functions
With $e^{u}$ and $\ln u$, the second derivative gets messy, so stay organized.
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…
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.
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…
Given the function $f(x) = \ln(x^2 + 9)$, determine:
-
The intervals where the function is concave ($\cap$) and where it is convex ($\cup$).
-
The coordinates of the points of inflection.
-
The equation of the tangent line to the graph of $f(x)$ at its point of inflection with a positive x-coordinate.
Rational Functions and Asymptotes
For a quotient, simplify $f'$ before differentiating again. Cancel common factors and the denominator stays small, so the algebra is much less error-prone. Then build a sign chart that includes both the zeros of $f''$ and the $x$-values where $f$ is undefined (vertical asymptotes).
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…
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.
-
Find $f''(x)$.
-
Study its sign in the corresponding intervals (bearing in mind that $x=1$ does not belong to the domain).
-
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.…
Given the rational function $f(x) = \frac{x^2-1}{x^2+3}$, carry out a detailed study of its curvature. To do this:
-
Find the intervals of concavity and convexity.
-
Determine the exact coordinates of its points of inflection, if they exist.
Given the function $f(x) = \frac{x^3}{x^2 + a}$, where $a$ is a nonzero real constant: a) Show that if $a > 0$, the function always has exactly three collinear points of…
Given the function $f(x) = \frac{x^3}{x^2 + a}$, where $a$ is a nonzero real constant:
- a) Show that if $a > 0$, the function always has exactly three collinear points of inflection.
- 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.
Finding Parameters from Inflection Conditions
Many problems give you a family of functions and a condition, and ask for the unknown constants. Each condition becomes one equation.
Slide $a$ and watch where the graph switches from concave down to concave up. For $f(x) = x^3 + ax^2 + 3x$, we have $f''(x) = 6x + 2a$, so the switch happens at $x = -\tfrac{a}{3}$.
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…
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 $x = 0$.
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…
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 $c$.
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…
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.
Piecewise Functions and Smooth Joins
When a function is defined in pieces, the junction at $x = c$ can be matched with different levels of smoothness.
| Requirement | Equation |
|---|---|
| Continuous at $c$ | $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$ |
| Differentiable at $c$ (same slope) | $f_{\text{left}}'(c) = f_{\text{right}}'(c)$ |
| Same curvature at $c$ | $f_{\text{left}}''(c) = f_{\text{right}}''(c)$ |
Each requirement gives one equation for the unknown constants. A point of inflection can also occur at the junction even when $f''(c)$ does not exist, as long as $f$ has a tangent line there and $f''$ has opposite signs on the two sides.
For a cubic piece $h(x) = ax^3 + bx^2 + cx + d$, remember $h'(x) = 3ax^2 + 2bx + c$ and $h''(x) = 6ax + 2b$. Four unknowns need four equations, such as value, slope, and second derivative matching at the junction, plus an inflection condition $h''(x_0) = 0$ elsewhere.
Let the piecewise function be: $$f(x) = \begin{cases} x^2 + 4x & \text{if } x \leq 0 \\ a \cdot \sin(x) + b \cdot x & \text{if } x > 0 \end{cases}$$ Calculate the values…
Let the piecewise function be:
$$f(x) = \begin{cases} x^2 + 4x & \text{if } x \leq 0 \\ a \cdot \sin(x) + b \cdot x & \text{if } x > 0 \end{cases}$$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.
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…
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 $x = 1$.
- a) Determine the values of $a$ and $b$.
- b) For the values found, study the intervals of concavity and convexity of the function and check if there are any additional inflection points.
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…
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 $x = 1$:
-
The functions match ($f$ is continuous).
-
The slopes match ($f$ is differentiable).
-
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 $d$.
Behavior at the Edges: Infinity and Domain Boundaries
Some problems ask what happens to curvature as $x \to \infty$ or as $x$ approaches the edge of the domain. Three ideas help:
-
Find the sign of $f''$ for large $x$. If it stays positive, the graph is concave up as $x \to \infty$.
-
Find $\lim f''(x)$. If it is $0$, the graph straightens out.
-
A boundary point of the domain (like $x = 0$ for $x > 0$) is not an inflection point, because the function is not defined on both sides.
For limits like $x^2 e^{-x^2}$ or $x^2 \ln x$, rewrite as a fraction and use L'Hôpital's Rule. Exponential functions grow faster than any power of $x$, so $e^{x^2}$ beats $x^2$ in the limit. In the same way, $\ln x$ grows more slowly than any positive power of $x$.
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…
Given the function $f(x) = x \cdot e^{-x^2}$, find:
-
The intervals of concavity and convexity.
-
The points of inflection of the curve.
-
What happens to the curvature of the function as $x \to \infty$? Reason your answer based on the limit of the second derivative.
Consider the function $f(x) = \frac{e^x}{x^2}$. a) Study the domain and the asymptotes of the function. b) Determine the increasing and decreasing intervals, and the…
Consider the function $f(x) = \frac{e^x}{x^2}$.
- a) Study the domain and the asymptotes of the function.
- b) Determine the increasing and decreasing intervals, and the relative extrema.
- c) Analyze the curvature and show that there is a single point of inflection. Determine its exact coordinates.
- 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 $x > 0$. a) Calculate $f'(x)$, $f''(x)$ and $f'''(x)$. b) Determine the intervals of concavity and convexity.…
Let the function $f(x) = x^4 \cdot \ln(x)$ be defined for $x > 0$.
- a) Calculate $f'(x)$, $f''(x)$ and $f'''(x)$.
- b) Determine the intervals of concavity and convexity.
- 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.
- 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?
Deciding the Sign of $f''$: Always Concave?
After simplifying, $f''$ is often a fraction with a positive denominator (a square or a positive power). Its numerator is a polynomial, so the question becomes: where is this polynomial positive or negative?
Back to the skateboard ramps from the start: the ramp that "steepens" is concave up ($f'' > 0$), the one that "flattens" is concave down ($f'' < 0$), and the spot where one design turns into the other is an inflection point, where $f''$ changes sign and the slope is at its steepest or flattest.
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…
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 $x \in [-2, 2]$.
- a) Calculate the value of $k$ so that the ramp changes its curvature (has a point of inflection) exactly at $x = 0$.
- 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.
Let the function be $f(x) = \text{arctg}(x^2 - 1)$. a) Find the expression for $f''(x)$ simplified as much as possible. b) Determine the points on the curve where the…
Let the function be $f(x) = \text{arctg}(x^2 - 1)$.
- a) Find the expression for $f''(x)$ simplified as much as possible.
- 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.
- c) Study the intervals of curvature of $f(x)$.