The first derivative tells you whether a graph is going up or down. The second derivative tells you how it bends. Two graphs can both be rising, one curving up more and more steeply and the other leveling off. That difference is concavity, and the points where the bending switches direction are inflection points. Together with the sign of
What Concavity Means
A function is concave up on an interval if its slope
Concavity is separate from increasing and decreasing. A concave up graph can be falling (the left side of a cup) or rising (the right side). What matters is whether the slope is getting bigger or smaller.
The Concavity Test
Since
Concavity test.
- If
on an interval, then is concave up on that interval. - If
on an interval, then is concave down on that interval.
This is the same idea as the first derivative test, moved up one level:
What Is a Point of Inflection?
A point of inflection is a point
If
- points where
, and - points where
is undefined, with in the domain of .
Being a candidate isn’t enough. You still have to check that
How to Find Concavity and Inflection Points
- Find
and simplify or factor it. - Find the candidates: where
or is undefined. Also note any points not in the domain. - Make a sign chart for
and test one point in each interval. - Read the concavity. + means concave up, − means concave down.
- Find the inflection points: candidates in the domain where the sign changes. Plug each into
to get the point.
Example 1: A Cubic
Find the intervals of concavity and the inflection point of
- Concave down on
, concave up on . - The concavity changes at
, so the inflection point is .
This is the same cubic as in the second derivative test lesson. Its local maximum at
Example 2: A Quartic With Two Inflection Points
Find the intervals of concavity and the inflection points of
The candidates are
| Interval | Test point | Concavity | |
|---|---|---|---|
| up | |||
| down | |||
| up |
The sign changes at both candidates, and
Example 3: f″ = 0 but No Inflection Point
Does
This is the most common mistake in this topic:
Example 4: An Inflection Point Where f″ Is Undefined
Find the inflection point of
Example 5: An Exponential Function
Find the intervals of concavity and the inflection point of
From the second derivative test lesson,
- Concave down on
, concave up on . - Inflection point:
.
Example 6: Concavity Changes, but No Inflection Point
Find the intervals of concavity of
Concavity From the Graph of f′ or f″
Exam questions often show a graph of a derivative instead of
- Given the graph of
: is concave up where is increasing and concave down where is decreasing. Inflection points of are where has a local max or min. - Given the graph of
: is concave up where the graph of is above the x-axis and concave down where it’s below. Inflection points are where it crosses the axis, not where it only touches.
For example, if the graph of
Concavity and the Second Derivative Test
The second derivative test is concavity applied at a critical point. If
In applications, an inflection point marks where a rate of change is fastest or slowest. In logistic growth, the population grows fastest at the inflection point, halfway to the carrying capacity. After it, growth keeps going but slows down: the point of diminishing returns.
Common Mistakes
- Calling every zero of
an inflection point. Check the sign change, as in Example 3. - Forgetting where
is undefined. Example 4 has an inflection point where doesn’t exist. - Calling a domain gap an inflection point. In Example 6, the concavity changes across an asymptote.
- Mixing up concavity and direction. Concave up doesn’t mean increasing.
- Giving only the x-value. An inflection point is a point
, though many questions ask only where it occurs.
Practice Problems
Try each one before opening the answer. The practice set has 12 more with instant checking.
Problem 1. Find the intervals of concavity and the inflection point of
Show answer
Answer: concave down on
Problem 2. Find the intervals of concavity and the inflection points of
Show answer
The sign is + on
Answer: concave up on
Problem 3. Find the intervals of concavity of
Show answer
Answer: concave down on
Problem 4. Find the inflection points of the bell curve
Show answer
By the chain rule,
The sign is the sign of
Answer:
Problem 5. Does
Show answer
Answer: no;
Problem 6. The derivative
Show answer
Answer: concave up on
Concavity and inflection points complete the toolkit for reading a graph from its derivatives. Next, optimization puts these tools to work on real problems: finding the largest area, the lowest cost or the shortest time.