Find the intervals where f is concave upwards and downwards, and inflection points (based on graph)

Question

It's easy to find the concavity based on an equation, but for some reason I don't understand how you could figure it out just by looking at it. I also need to find the inflection points in a certain order. What can I do to figure it out, and could you provide one example?

Answer 1

Intuitively, there are a couple ways you could determine the concavity of a graph by looking at it:

1. If the graph is opening "up" then it's concave up. Imagine connecting the top two graphs in the image below. You should get an upward-shaped parabola. Conversely, if the graph is opening "down" then it's concave down. Connect the bottom two graphs and you should get a downward-shaped parabola.
2. You can also determine the concavity of a graph by imagining its tangent lines. If all the tangent lines are below the graph, then it's concave up. If all the tangent lines are above the graph, then it's concave down. If the tangent line is on the actual graph, then you have an inflection point (i.e. a straight line). The picture below demonstrates this.

Now let's consider your problem. From $0<x<2$ the graph is opening upwards (if we were to continue drawing this segment, it would look like an upward-shaped parabola). So the graph is concave up in the interval $0<x<2$.

From $2<x<3$ the graph is opening downwards. So the graph is concave down in the interval $2<x<3$.

For a smooth graph (do you know what this means?) an inflection point always lies between concave up and concave down segments. So one inflection point would be $x=2$ based in our inferences above.

Can you proceed in a similar manner?