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Figure 9.5

ogy, a function whose graph is strictly concave (strictly convex) is called a strictly concave (strictly convex) function. The precise geometric characterization of a strictly concave function is as follows. If we pick any pair of points M and N on its curve and join them by a straight line, the line segment MN must lie entirely below the curve, except at points M and N. The characterization of a strictly convex function can be obtained by substituting the word "above" for the word "below" in the last statement. Try this out in Fig. 9.5. If the characterizing condition is relaxed somewhat, so that the line segment MN is allowed to lie either below the curve, or along (coinciding with) the curve, then we will be describing instead a concave function, without the adverb "strictly." Similarly, if the line segment MN either lies above, or lies along the curve, then the function is convex, again without the adverb "strictly." Note that, since the line segment MN may coincide with a (nonstrictly) concave or convex curve, the latter may very well contain a linear segment. In contrast, a strictly concave or convex curve can never contain a linear segment anywhere. It follows that while a strictly concave (convex) function is automatically a concave (convex) function, the converse is not true.*

From our earlier discussion of the second derivative, we may now infer that if the second derivative f"(x) is negative for all a, then the primitive function f(x) must be a strictly concave function. Similarly, f(x) must be strictly convex, if f"(x) is positive for all x. Despite this, it is not valid to reverse the above inference and say that, if f(x) is strictly concave (strictly convex), then f"(x) must be negative (positive) for all x. This is because, in certain exceptional cases, the second derivative may have a zero value at a stationary point on such a curve. An example of this can be found in the functiony = f(x) = x4, which plots as a strictly convex curve, but whose derivatives

* We shall discuss these concepts further in Sec. 11.5 below.

indicate that, at the stationary point where x = 0, the value of the second derivative is /"(0) = 0. Note, however, that at any other point, with x =j= 0, the second derivative of this function does have the (expected) positive sign. Aside from the possibility of a zero value at a stationary point, therefore, the second derivative of a strictly concave or convex function may be expected in general to adhere to a single algebraic sign.

For other types of function, the second derivative may take both positive and negative values, depending on the value of x. In Fig. 9.3a and b, for instance, both /(x) and g(x) undergo a sign change in the second derivative at their respective inflection points J and K. According to Fig. 9.3a', the slope of /'(x)—that is, the value of /"(x)—changes from negative to positive at x = /; the exact opposite occurs with the slope of g'(x)—that is, the value of g"(x)—on the basis of Fig. 9.3b'. Translated into curvature terms, this means that the graph of /(x) turns from concave to convex at point J, whereas the graph of g(x) has the reverse change at point K. Consequently, instead of characterizing an inflection point as a point where the first derivative reaches an extreme value, we may alternatively characterize it as a point where the function undergoes a change in curvature or a change in the sign of its second derivative.

An Application

The two curves in Fig. 9.5 exemplify the graphs of quadratic functions, which may be expressed generally in the form y = ax2 + bx + c (a + 0)

From our discussion of the second derivative, we can now derive a convenient way of determining whether a given quadratic function will have a strictly convex (U-shaped) or a strictly concave (inverse U-shaped) graph.

Since the second derivative of the quadratic function cited is d2y/dx2 = 2a, this derivative will always have the same algebraic sign as the coefficient a. Recalling that a positive second derivative implies a strictly convex curve, we can infer that a positive coefficient a in the above quadratic function gives rise to a U-shaped graph. In contrast, a negative coefficient a leads to a strictly concave curve, shaped like an inverted U.

As intimated at the end of Sec. 9.2, the relative extremum of this function will also prove to be its absolute extremum, because in a quadratic function there can be found only a single valley or peak, evident in a U or inverted U, respectively.

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