Use the given graph of 𝑓 to find the coordinates of the points of inflection.
Well, to understand how to solve this problem, we need to know what are the points of inflection. Well, the points of inflection are actually the points where there is a change of concavity. So what I actually mean by a change of concavity is where the shape of our function changes. So it can change from concave down to concave up or concave up to concave down.
So in order to find our points of inflection, we’re gonna have a look at our graph and actually see where it’s concave up and concave down. Well, we can actually see in this first section our function is actually concave down. And then, we change to a section which I’ve marked in pink, where we can actually see that the function is concave up. And then, we have another section of marks in orange, which is a short section of concave down. And then, finally, the last section of our function in the graph is actually concave up.
Okay, great, so we now marked on the concave down and concave up segments of our function. So this is where we go back to our definition because we now want to find the points of inflection. And we can see that actually the points of inflection are the points, where the concavity actually changes. And I’ve marked on with red crosses where the concavity changes. You can see we go from concave down to concave up and then concave up to concave down and back from concave down to concave up.
So therefore, we can actually say that the points of inflection are the first point two, two and that’s because we change from concave down to concave up. Our second point is four, three and that’s because we change from concave up to concave down. And then, our last change in concavity is at our final point five, four. So therefore, there are three points of inflection.