Here we examine what the second derivative tells us about the geometry of functions.

So far, we know how to determine if a function is increasing or decreasing by looking at the sign of its derivative. If all we know about \(f\) is the increasing/decreasing information, we no not have enough information to say how \(f\) is behaving accurately enough. Consider the following four possibilities:

In both graphs in the left-hand column, \(f\) is decreasing. In both graphs in the right-hand column, \(f\) is increasing. What is the difference between the two rows? Their concavity.

That is, the graph of \(f\) is concave up if the graph locally lies above its tangent lines, and is concave down if the graph locally lies below its tangent lines.

The graphs of two functions, \(f\) and \(g\), both increasing on the given interval, are given below.

In that example, examine the intervals where \(f\) was concave up. At the beginning of that interval, \(f\) was decreasing, and at the end, \(f\) was increasing. The slopes in that interval increased from a negative value to a positive value. In the interval where \(f\) was concave down, the slopes started positive and ended negative. The slopes in that interval were decreasing.

We know that the sign of the derivative tells us whether a function is increasing or decreasing at some point. Likewise, the sign of the second derivative \(f''(x)\) tells us whether \(f'(x)\) is increasing or decreasing at \(x\). Let’s use this to add more details into the chart from above.

If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. It is worth summarizing what we have seen already in to a single theorem.

Let’s start with information about where a function is increasing/decreasing/concave up/concave down, and see if we can use that to understand the behavior of the function.