Locating extrema Consider the graph of a function ƒ on the int...

Question

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3].

c. Give the approximate coordinates of the inflection point(s) of f.

Accepted Answer

Hello. In this video, we are going to be identifying the values of X that represent an inflection point. We are told that we are given a function F of X defined on the interval from -3 to 5.5. Which of the following X values represents an inflection point of F of X? Now, in order to identify this point, first, let's go ahead and quickly review what it means to be an inflection point. First, when we are talking about inflection points, we need to understand the idea of concavity. Concavity is a characteristic of the graph that represents either concave up or concave down. The easiest way to distinguish the two is to imagine a cup shape. If a graph is concave down, then you can imagine the behaviors of the graph to be an upside down cup. And vice versa, if the graph is concave up, then you can imagine the behaviors of the graph to be an upside upright cup. Now, for inflection point, an inflection point is located when the concavity of a graph switches. So, for example, let's say we have a behavior that is concave up, then at a point, the concavity switches to be concave down, or vice versa, we have a concave down shape, which eventually switches to be concave up. This is what it means to be an inflection point. So, where do we see a switch of concavity on the given graph? Well, if we take a look at the graph, at the point, or I'm sorry, at the endpoint -3, going up to the local maximum -1, we have a concave up or concave down shape. It'll be concave down going from -3 to -1. Now, at the X value, X is equal to -1, the graph is still concave down, but notice that at the X value, X is equal to 1, the concavity switches from concave down to concave up. The graph is concave up from the X value 1 to 3. So what this means is that at the value X is equal to 1, we have a shift in concavity, which means that X is equal to 1 is going to be the inflection point of the graph. So, I hope this video helps you in understanding how to identify inflection points, and I will go ahead and see you all in the next video.

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
c. Give the approximate coordinates of the inflection point(s) of f.

Key Concepts

Inflection Points

Second Derivative Test

Graphical Analysis

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