Lesson 9 of 12, 12 min read
Concavity and inflection points
Using the second derivative to find where a graph is concave up or down, locating inflection points, and the second derivative test for relative extrema.
How a graph bends
Two functions can both be increasing yet bend differently: \(x^2\) (for \(x > 0\)) rises faster and faster, while \(\sqrt{x}\) rises more and more slowly. The difference is concavity.
The second derivative test for concavity
When a graph is concave up, the tangent slopes increase as you move right, so \(f'\) is increasing and \(f'' > 0\).
Inflection points
The second derivative test for extrema
Practice
Quick check
If \(f''(x) < 0\) on an interval, the graph is:
A negative second derivative means the slopes are decreasing.
Quick check
\(f'(4) = 0\) and \(f''(4) = -3\). What happens at \(x = 4\)?
Horizontal tangent and concave down: a maximum.
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