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We discuss how to find points of inflection of a function using the second derivative.
Video Lecture
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Text and Additional Details
We’ve seen what inflection points are, but as usual we want to know how to find these points rigorously, that is, how to solve for them
algebraically. In this segment we aim to do just this by developing the mechanics and results necessary to find, and verify, inflection
points.
We’ve seen that inflection points represent transition points between concavity. The easiest way to think of inflection points then, is to
consider the parallel between them and local extrema which we’ve already discussed.
Recall that local extrema occur at critical points, or more specifically, they occur at points where the first derivative transitions - either
from negative to positive or positive to negative. But this is essentially the same as what we are after for inflection points, which occur
when the second derivative similarly transitions - either from positive to negative, or negative to positive. This means that the process for
finding inflection points is essentially identical to the process of finding local extrema, just using the second derivative rather than the
first!
With this idea established, we can formally state the definition of inflection point:
Inflection Point Let be a second differentiable function. Fix such that and suppose . Then is called an inflection point if the
curvature of the function changes sign across .
In normal human-speak what the above says is that a point is an inflection point if the concavity changes at , from concave up to concave
down, or the other way around.
The process of finding inflection points is essentially the same as...
Finding where a function is increasing.Finding and classifying
local extrema.Finding the zeros of a function.Destroying the one ring - it’s an arduous trek full of mortal danger.
Formal definition aside, what we really care about is how to find these points. As mentioned, the process of doing so is the same as finding
local extrema, but using the second derivative instead of the first. In particular, we start by computing, and then finding the zeros and
domain restrictions, of the second derivative. Once we have found all the zeros and the domain restrictions we form a
sign chart for the second derivative function. Anywhere that the sign changes on either side of a zero of (according to
the sign chart), is then an inflection point. We will have numerous videos with concrete functions to show this process
explicitly.
In this segment we discussed the formal definition of inflection points, as well as the general approach to finding them. Actual concrete
examples will be covered in separate example videos, but as we discussed, the process is nearly identical to finding the local extrema, just
applied to the second derivative function rather than the first.
Business Calculus One