# Does the second derivative test tell you anything about inflection points?

I am confused by two sources of information. Wikipedia tells me that the second derivative test cannot be used to determine inflection points. However, on the Harvey Mudd calculus page, it says that you can:

"If f'(c) exists and f''(c) changes sign at x=c, then the point (c,f(c)) is an inflection point of the graph of f. If f''(c) exists at the inflection point, then f''(c)=0"

http://www.math.hmc.edu/calculus/tutorials/secondderiv/

My book doesn't mention anything about it, but I learned from my instructor that you could use the second derivative test to find inflection points. Could anyone clear up this disagreement in calculus?

## 2 Answers

An inflection point is an extremum of the first derivative (if that exists), so maybe you can take it from there. Recall that not every point where the first derivative is zero is an extremum (nor does the first derivative necessarily exist at an extremum).

The second derivative test uses the sign of the second derivative at a critical point to determine if the critical value is a local minimum (second derivative positive there) or maximum (second derivative negative there).

If the second derivative is actually zero there, you can't tell if it is a local minimum, local maximum, or neither (the second derivative test gives no result). But in this case, the critical point is a candidate for an inflection point; it will be an inflection point if the second derivative is not only zero there, but actually changes sign there. Note that it could be zero there but also be positive on both sides (or negative on both sides), in which case it is not an inflection point.

The second derivative describes how a curve bends locally. An inflection point is where the curve bends down from the tangent line on one side and up from the tangent line on the other side.

The second derivative test relies on the observation that if the tangent is horizontal at a point and the curve bends upward from the tangent at that point, the point is a local minimum (ditto for local max, mutandis mutatis).

Think about the shape of the curve $y = \sin x$ at the origin. Draw the curve and its tangent there, and observe the sign of the second derivative near the origin.