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Concavity, Points of Inflection, and the Second Derivative TestConcavity and Points of Inflection
When a curve is concave up, it is sort of bowlshaped, and you can think it might hold water. When it is concave down, it is sort of upsidedownbowllike, and water would run off of it. HowtoThe intervals of concavity can be found in the same way used to determine the intervals of increase/decrease, except that we use the second derivative instead of the first. In particular, since $(f')'=f''$, the intervals of increase/decrease for the first derivative will determine the concavity of $f$:
The Second Derivative Test
The reasoning behind the test is simple: if $f''(c) \gt 0$, then $f'(x)$ is increasing near $x=c$. Since $f'(c)=0$, this means that $f'(x)$ used to be negative and is about to be positive. So the curve bottoms out at $x=c$ and then heads back up. The critical number $x=c$ is the bottom of the concaveup bowl. Likewise, if $f''(c) \lt 0$ and $f'(c)=0$, then $f'(x)$ is decreasing; it used to be positive and is about to be negative. The point $x=c$ is at the top of an upsidedown bowl. Inflection PointsAn inflection point is a point where concavity changes sign from plus to minus or from minus to plus.
Visual Wrapup
