Consider the problem of maximizing a differentiable function f(x) of a single unconstrained variable x. Let x 0 and x0, respectively, be a valid lower bound and upper bound on the same global maximum (if one exists). Prove the following general properties of the one-dimensional search procedure (as presented in Sec.

13.4) for attempting to solve such a problem.

(a) Given x 0, x0, and  0, the sequence of trial solutions selected by the midpoint rule must converge to a limiting solution.
[Hint: First show that limn(x n  x n) 0, where xn and x n are the upper and lower bounds identified at iteration n.]

(b) If f(x) is concave [so that df(x)/dx is a monotone decreasing function of x], then the limiting solution in part

(a) must be a global maximum.

(c) If f(x) is not concave everywhere, but would be concave if its domain were restricted to the interval between x 0 and x 0, then the limiting solution in part

(a) must be a global maximum.

(d) If f(x) is not concave even over the interval between x 0 and x 0, then the limiting solution in part

(a) need not be a global maximum. (Prove this by graphically constructing a counterexample.)

(e) If df(x)/dx  0 for all x, then no x 0 exists. If df(x)/dx 0 for all x, then no x 0 exists. In either case, f(x) does not possess a global maximum.

(f) If f(x) is concave and lim x f(x)/dx  0, then no x 0 exists. If f(x) is concave and limx df(x)/dx 0, then no x 0 exists. In either case, f(x) does not possess a global maximum.