Concavity
In short: Concavity describes which way a curve bends: concave up holds water like a cup and means f” is positive, while concave down spills it like a cap and means f” is negative.
AP Calculus ABCalculusAP Calculus AB Unit 5
What is Concavity?
Concavity describes the direction in which a graph bends. Where the curve is shaped like a cup, tangent lines lie below it and the slopes keep getting larger. Where it is shaped like a cap, tangent lines lie above it and the slopes keep getting smaller.
Because the second derivative measures how the slope itself is changing, \(f^{\prime\prime}(x) > 0\) on an interval means the graph is concave up there, and \(f^{\prime\prime}(x) < 0\) means it is concave down. The input where the bending direction flips is an inflection point.
How to spot it
- A question asks on which intervals a graph is concave up or concave down.
- The second derivative is given, or you are asked to compute it and check its sign.
- A rate story says something is speeding up its growth, or growing more slowly, as time passes.
- A graph looks like a bowl in one region and an upside-down bowl in another.
The method
- Differentiate twice to get \(f^{\prime\prime}(x)\).
- Find where \(f^{\prime\prime}(x) = 0\) or is undefined; these inputs cut the number line into intervals.
- Test one value from each interval in \(f^{\prime\prime}\) and record the sign.
- Positive means concave up on that interval; negative means concave down.
Concavity: solved example
Problem
Determine where \(f(x) = 2x^3 – 9x^2 + 12x – 5\) is concave up and where it is concave down.
Solution
\(f^{\prime}(x) = 6x^2 – 18x + 12\), so \(f^{\prime\prime}(x) = 12x – 18\). It equals zero at \(x = \frac{3}{2}\).
Test \(x = 0\): \(f^{\prime\prime}(0) = -18 < 0\). Test \(x = 2\): \(f^{\prime\prime}(2) = 6 > 0\).
So the graph is concave down on \(\left(-\infty, \frac{3}{2}\right)\) and concave up on \(\left(\frac{3}{2}, \infty\right)\).
Try it
Question 1
Is \(g(x) = x^4\) concave up or down at \(x = -1\)?
Show the answer
\(g^{\prime\prime}(x) = 12x^2\), so \(g^{\prime\prime}(-1) = 12 > 0\): concave up.
Question 2
A town’s population satisfies \(P^{\prime}(t) > 0\) and \(P^{\prime\prime}(t) < 0\). Describe the growth.
Show the answer
The population is increasing, but the graph is concave down, so the growth is slowing.
Concavity vs. Inflection Point
Concavity is a property of an entire interval: the graph is a cup or a cap there. An inflection point is a single location where one kind of bending turns into the other. You can describe concavity without ever finding an inflection point, for instance on \(x^2\), which is a cup everywhere.
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