Modify the function in Example 8.1 so that it becomes f(x) = (x -1)(x -2)(x -3)(x - 4)(x - 5)(x - 6)(x - 7) for 1  x  7. Plot a lot of points from 1 to 7 to see what the graph of this function looks...

Modify the function in Example 8.1 so that it becomes f(x) = (x -1)(x -2)(x -3)(x - 4)(x - 5)(x - 6)(x - 7) for 1
 x
 7. Plot a lot of points from 1 to 7 to see what the graph of this function looks like. Then use GRG Nonlinear Solver to find its maximum. Try the following starting points (and don’t use the Multistart option): 1, 3, 5, 6, and 6.9. Report what you find. Then try Evolutionary Solver. Does it find the correct solution?

EXAMPLE 8.1 MAXIMIZING A NONLINEAR FUNCTION WITH LOCAL MAXIMA

To see how Evolutionary Solver works, we consider a simple function that is difficult for GRG Nonlinear Solver. This example, analyzed in Chapter 7, is a function of a single variable x: f(x) = (x - 1)(x - 2)(x - 3)(x - 4)(x - 5) for 1
 x
 5. The objective is to maximize f(x) over this range. However, the graph of this function shown in Figure 8.2 indicates that there are two local maxima: one at around x = 3.5 and the other at x = 5. The global maximum, the one we want, is near x = 1.5. Can Evolutionary Solver find this global maximum?

Figure 8.2 Function with Local Maxima

Objective To illustrate how Evolutionary Solver works and to see how it can successfully find a global maximum for a function with several local maxima.