There is an alternative type of definition that helps in the interpretation of the second order derivative.
The important idea is that we can use the features of polynomials as a standard to measure the features of other functions. For example the tangent line can be considered the closest line to a graph. The closest line to the graph is the best linear approximation to the graph at the point of tangency. Being the closest line the direction of the tangent line is also the direction of the graph. The slope of the tangent line will then tell us whether the graph is increasing or decreasing at the point of tangency.
While direction characterizes lines, parabolas can be characterized by their concavity. The tangent parabola is the closest parabola to the graph at a given point or, again, the best quadratic approximation to the graph. The second derivative is, within a factor, the coefficient of the x^2 term of the tangent parabola. That coefficient determines whether the parabola is concave up or concave down.
More than two hundred year ago Roger Joseph Boscovich made an interesting observation on this topic.
"We fashion our geometry on the properties of a straight line because that seems the simplest of all. But in reality all lines that are continuous and of a uniform nature are just as simple as one another. Another kind of mind which might form an equally clear mental perception of some property of any one of these curves, as we do the congruence of a straight line, might believe these curves to be the simplest of all, and from that property of these curves build up the elements of a very different geometry, referring all other curves to that one, just as we compare them to a straight line. Indeed, these minds, if they noticed and formed an extremely clear perception of some property of say, the parabola, would not seek, as our geometers do, to rectify the parabola, they would endeavor, if one may coin the expression, to parabolify the straight line."
The second derivative "parabolifies" a graph.
In a message dated 11/12/2008 9:59:54 A.M. Eastern Standard Time, firstname.lastname@example.org writes:
I understand why the first derivative test tells you where the function is increasing or decreasing but why do we use the second derivative test to tell us where the function is concave up or concave down?