Page 118 - Calculus Workbook For Dummies

102 Part III: Differentiation

Smiles and Frowns: Concavity

and Inflection Points

Another purpose of the second derivative is to analyze concavity and points of inflec-
tion. (For a refresher, look at Figure 7-3: The area between A and B is concave down —
like an upside-down spoon or a frown; the areas on the outsides of A and B are con-
cave up — right-side up spoon or a smile; and A and B are inflection points.) A positive
second derivative means concave up; a negative second derivative means concave
down. Where the concavity switches from up to down or down to up, you’ve got an
inflection point, and the second derivative there will be zero (or sometimes undefined).

A

B
Figure 7-3:
Concavity
and points
of inflection.

All inflection points have a second derivative of zero (if the second derivative exists),
but not all points with a second derivative of zero are inflection points. This is no dif-
ferent from “all ships are boats but not all boats are ships.”

However, you can have an inflection point where the second derivative is undefined.
This occurs when the inflection point has a vertical tangent and in some bizarre
curves that you shouldn’t worry about that have a weird discontinuity in the second
derivative.