Chapter 16: Concavity and Inflection Points
Beginner
16.1 Definition of Concavity
- Concave up (convex function): If $f''(x)>0$ on an interval $I$, the graph is concave up on that interval, and the tangent line at each point lies below the graph.
- Concave down (concave function): If $f''(x)<0$ on an interval $I$, the graph is concave down on that interval, and the tangent line at each point lies above the graph.
16.2 Inflection Points
A point where the concavity of a curve changes is called an inflection point.
An inflection point is a point where the concavity of a curve changes across $x=a$. When $f''$ can be evaluated on both sides of $x=a$, check whether its sign changes from one side to the other. A point where $f''(a)=0$ is a candidate for an inflection point, but this condition alone does not guarantee that it is one. There are also inflection points where $f''(a)$ does not exist.
$f'(x) = 3x^2$, $f''(x) = 6x$
Setting $f''(x) = 0$ gives $x = 0$.
For $x < 0$, $f'' < 0$ (concave down); for $x > 0$, $f'' > 0$ (concave up).
Therefore, $x = 0$ is an inflection point.
16.3 Sketching the Graph
Derivatives
$f'(x) = 4x^3 - 12x^2 = 4x^2(x-3)$
$f''(x) = 12x^2 - 24x = 12x(x-2)$
Local Extrema
Setting $f'(x) = 0$ gives $x = 0, 3$.
At $x = 0$: $f''(0) = 0$ (the second derivative test is inconclusive; use the first derivative test).
At $x = 3$: $f''(3) = 36 > 0$ → local minimum, $f(3) = 81 - 108 + 6 = -21$.
Inflection Points
Setting $f''(x) = 0$ gives $x = 0, 2$.
The sign of $f''$ changes at both points, so both are inflection points.
Inflection points: $(0, 6)$ and $(2, -10)$.
Frequently Asked Questions
What is the intuitive meaning of concave up and concave down?
For a concave-up graph, the line segment (chord) connecting any two points on the graph lies above the graph. The opposite is true for a concave-down graph. A graph is concave up on an interval where $f''(x)>0$ and concave down where $f''(x)<0$.
What is an inflection point, and how do you find one?
An inflection point is a point where concavity changes. When $f''$ can be evaluated on both sides of $x=a$, check whether its sign changes across that point. If $f''(a)$ exists at an inflection point, then $f''(a)=0$, but the converse is not necessarily true ($x=0$ for $f(x)=x^4$ is not an inflection point even though $f''(0)=0$). An inflection point can also exist where $f''(a)$ does not exist. For example, $x=0$ for $f(x)=x^3$ is an inflection point.
What is the difference between an inflection point and a local extremum?
An inflection point is where concavity changes (the sign of $f''$ changes), whereas a local extremum is where increasing or decreasing behavior changes (the sign of $f'$ changes). These are different concepts: an inflection point concerns a change in concavity, while a local extremum concerns a change in monotonicity. They generally do not coincide. For $f(x) = x^3$, $x = 0$ is an inflection point but not a local extremum.
Exercises
Find the inflection point of $f(x) = x^3 - 6x^2 + 9x + 1$.
Show Answer
Setting $f''(x) = 6x - 12 = 0$ gives $x = 2$.
For $x < 2$, $f'' < 0$; for $x > 2$, $f'' > 0$.
Inflection point: $(2, f(2)) = (2, 3)$.