PRELIMINARY CALCULUS
G
f(x)xFigure 2.3 The graph of a functionf(x) that has a general point of inflection
at the pointG.Now, we examine each stationary point in turn. Forx=3,d^2 f/dx^2 = 30. Since this is
positive, we conclude thatx= 3 is a minimum. Similarly, forx=−2,d^2 f/dx^2 =−30 and
sox=−2 is a maximum.
So far we have concentrated on stationary points, which are defined to havedf/dx= 0. We have found that at a stationary point of inflectiond^2 f/dx^2 is
also zero and changes sign. This naturally leads us to consider points at which
d^2 f/dx^2 is zero and changes sign but at whichdf/dxisnot, in general, zero. Such
points are calledgeneral points of inflectionor simplypoints of inflection. Clearly,
a stationary point of inflection is a special case for whichdf/dxis also zero.
At a general point of inflection the graph of the function changes from being
concave upwards to concave downwards (or vice versa), but the tangent to the
curve at this point need not be horizontal. A typical example of a general point
of inflection is shown in figure 2.3.
The determination of the stationary points of a function, together with theidentification of its zeros, infinities and possible asymptotes, is usually sufficient
to enable a graph of the function showing most of its significant features to be
sketched. Some examples for the reader to try are included in the exercises at the
end of this chapter.
2.1.9 Curvature of a functionIn the previous section we saw that at a point of inflection of the function
f(x), the second derivatived^2 f/dx^2 changes sign and passes through zero. The
corresponding graph offshows an inversion of its curvature at the point of
inflection. We now develop a more quantitative measure of the curvature of a
function (or its graph), which is applicable at general points and not just in the
neighbourhood of a point of inflection.
As in figure 2.1, letθbe the angle made with thex-axis by the tangent at a