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where a function increases and decreases is not sufficient to draw a good graph. In
Figures 4.53a and 4.53b, we show two very different shapes of increasing functions
joining the same two points.
y
a
FIGURE 4.53a b x y a FIGURE 4.53b b x
Increasing function Increasing function
Note that the rate of growth in Figure 4.53a, is increasing, while the rate of growth
depicted in Figure 4.53b is decreasing. As a further illustration of this, Figures 4.54a
and 4.54b are the same as Figures 4.53a and 4.53b, respectively, but with a few tangent
lines drawn in.
y y

x x
a b a b

FIGURE 4.54a FIGURE 4.54b
Concave up, increasing Concave down, increasing

′
Although all of the tangent lines have positive slope [since f (x) > 0], the slopes of
the tangent lines in Figure 4.54a are increasing, while those in Figure 4.54b are decreas-
ing. We call the graph in Figure 4.54a concave up and the graph in Figure 4.54b concave
down. The situation is similar for decreasing functions. In Figures 4.55a and 4.55b, we

y y

x x
a b a b

FIGURE 4.55a FIGURE 4.55b
Concave up, decreasing Concave down, decreasing

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270 | Lesson 4-5 | Concavity and the Second Derivative Test