Question 10.3: Consider the objective function f(C) = e^–c cos C defined on......
Consider the objective function\ f(C) = e^{–c} cos C defined on the interval 0 ≤ C ≤ 15. It is clear that this function is infinitely differentiable for all real C, so its concavity is well defined on [0, 15]. In particular, the concavity of such an analytic function is completely characterized by its zeroth, first, and second derivatives. This is shown in Table 10.1. Characterize the convexity of f(C).
TABLE 10.1 Concavity Characteristics of the Objective Function f(C) as Defined by Derivative Properties
\ \begin{array}{c} \hline &\text{Positive}& \text{Zero}& \text{Negative}\\\hline f(C)& \text{Positive}& \text{Crossing point}& \text{Negative}\\f'(C)& \text{Increasing}& \text{Critical point}& \text{Decreasing}\\f″(C)& \text{Concave up}& \text{Inflection point}& \text{Concave down}\\\hline \end{array}
The derivatives of f(C) are straightforward. Noting that the “zeroth” derivative of a function is the function itself,
\ f(C) = e^{–c} cos C,\\f'(C)=-e^{-C}(cos C+ sin C),\\f″(C)=2e^{-C} sin C.
Since\ e^{-C} > 0 for all real C, the objective f(C) is completely characterized by the positiveness or negativeness of sin C. Crossing points on the constraint interval [0, 15] are at\ C_{crossing} =\frac{1}{2}\pi, \frac{3}{2}\pi, \frac{5}{2}\pi, and \frac{7}{2}\pi . In between each subinterval, the function goes from being positive on\ [0, \frac{1}{2}\pi] , negative on\ [\frac{1}{2}\pi, \frac{3}{2}\pi] , and alternating thereafter. This is shown graphically in Figure 10.7
Similarly, the critical points, which are candidates for local maxima and minima, are where the derivative is zero. Again ignoring the exponential term, the critical points are at\ C_{critical} = \frac{3}{4}\pi, \frac{7}{4}\pi, \frac{11}{4}\pi, and \frac{15}{4}\pi . On the subintervals, the function is either increasing or decreasing, as shown in Figure 10.7. The concavity is given where the second derivative is zero. As above, this is at\ C_{inflection} = 0, \pi, 2\pi, 3\pi, and 4\pi . As indicated in Figure 10.7, f(C) is either concave up or concave down on alternate intervals between the inflection points.
