Concavity and Inflection Points - Concept
Inflection points are points on the graph where the concavity changes. A positive second derivative means a function is concave up, and a negative second derivative means the function is concave down. These inflection points are places where the second derivative is zero, and the function changes from concave up to concave down or vice versa.
I want to talk about a new concept called "concavity." Now concavity describes the curvature of the graph of a function. I have 4 kinds of graphs here, the first two are both concave up but this is an example of a graph that's concave up and decreasing, this is an example of a graph that's concave up and increasing and if you're confused about what identifies this as concave up, you can draw some tangent lines, if you draw a tangent line you'll find, then you extend them you'll find that the tangent lines lie below the graph, the concave up graph lies above them. And the same thing whether the graph is decreasing or increasing so a concave up graph lies above its tangent lines.
Now if a graph is concave down it'll do the opposite, the tangent lines will be above the graph so both of these cases the curve lies below the tangent lines that's concave down. Now a concave up or a concave down graph can be increasing or decreasing and that's why we get these 4 different kinds of curves. Now what you'll need to know on your homework is how to test for a concavity and concavity has to do with the second derivative, if you think about it each of these tangent lines its slope comes from the first derivative and in this picture the slope, these all look negative but the slope would be actually increasing like say this is negative 5 this might be negative 2, negative a half the slope is actually increasing and the same thing is happening here maybe this is a slope of one third, 1, 2 this slope is also increasing. So when a graph is concave up the slope of the tangent line is increasing that means f prime is increasing f double prime is positive.
Now when a graph is concave down the opposite is happening, the slope of tangent line is decreasing like this might be negative 1, this might be negative 2, negative 4 and over here you've got positive slopes like maybe this is 5, this is 1, this is not very well drawn this is maybe one quarter. The slopes are decreasing here and so you'd say that f prime is decreasing or f double prime is negative and that's where the concavity test comes in. If you have a function f and i is some interval in its domain. If f double prime is positive on an interval for all x in that interval. Then f is concave up on that interval and if f double prime is negative, then f is concave down. So whether or not a function is concave up or concave down depends on whether the second derivative is positive or negative.
Positive means concave up negative means concave down, now there's one other term I need to tell you about and that's inflection point. It is possible for a graph to change concavity like for example something like this, here the graph is changing from concave down to concave up the point where that happens is called an inflection point. It's a point in the graph where the concavity changes and the way you check for that is you look for changes and sign of the second derivative. So again if the second derivative is positive the graph is concave up where it's negative the graph is concave down and when that sign changes you have an inflection point.