Calculus: Analytic Geometry and Calculus, with Vectors

5.3 Second derivatives, convexity, and flexpoints 305

The information contained in these theorems is sometimesvery helpful

when graphs of given functions f are being drawn. For example, Figure

5.34 shows an application of the first part of this theorem;f"(x) is positive

Slop! increasing

f '(x)> 0

Figure 5.34 Figure 5.35

over an interval and f' (x), the slope, increases from -1 through 0 to +1

as x increases over the interval. Figure 5.35 shows an application of the

second part of the theorem; f"(x) is negativeover an interval and f'(x),

the slope, decreases from 1 through 0 to -1 as x increases over the

interval. Sometimes it is helpful to put ideas involving derivatives in the

form

(5.351)

d2y- d dy _dm_d slope

dx2 dx(dx dx dx

The important thing to remember is that f"(x) is the derivative of f'(x)
and that a positive second derivative implies an increasing first derivative
and hence an increasing slope, and that a negative second derivative implies
a decreasing first derivative and hence a decreasing slope. It is sometimes
useful and even necessary to know about attempts to describe the dif-
ferences between the arcs of Figures 5.34 and 5.35 in other words. The
first runs through a depression and the second runsover a hump. The
first bends upward and the second bends downward. The first isconvex
upward and the second is convex t downward. In the first case, the
chord joining two points on the graph lies above thearc joining the two
points, and in the second case the chord lies below the arc. In the first
case each tangent to the graph lies (at least locally) below the arc, and
in the second case each tangent lies (at least locally) above thearc.
The virtue of the following theorem lies in the fact that it is a "local
theorem" which we can apply without determining signs of functions
over whole intervals and which is therefore sometimes easier to apply
than Theorem 5.28.
Theorem 5.36 If f'(xo) = 0 and f"(xo) > 0, then (as Figure 5.34

indicates) f has a local minimum at xo. If f'(xo) = 0 and f"(xo) < 0,
then (as Figure 5.35 indicates) f has a local maximum at xo.

f In mathematics and optics a point set (which might in some cases be a lens) is convex
if it contains the line segment joining P, and P2 whenever it contains P, and P_. The set
is sometimes said to be concave if it is not convex. When we say that a part of a graph in the
xy plane is convex upward, we mean that the set lying above it is convex; ue do not mean
that the graph is a convex set.