Points of Inflection Analytic View

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We discuss how to find points of inflection of a function using the second derivative.

Video Lecture

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 $f:[a,b]\rightarrow \mathbb {R}$ be a second differentiable function. Fix $c$ such that $a < c < b$ and suppose $f''(c) = 0$ . Then $c$ is called an inflection point if the curvature of the function changes sign across $c$ .

In normal human-speak what the above says is that a point $c$ is an inflection point if the concavity changes at $c$ , from concave up to concave down, or the other way around.

1 : 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 $f''(x)$ (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.