1. Consider the following nonconvex programming problem: Maximize f (x) _ 1,000x _ 400×2 40×3 _ x4, subject to
x2 x _ 500
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x _ 0.
(a) Identify the feasible values for x. Obtain general expressions for the first three derivatives of f (x). Use this information to
help you draw a rough sketch of f (x) over the feasible region for x. Without calculating their values, mark the points on your graph that correspond to local maxima and minima.
(b) Use the one-dimensional search procedure with _ _ 0.05 to find each of the local maxima. Use your sketch from part (a) to identify appropriate initial bounds for each of these searches. Which of the local maxima is a global maximum?
(c) Use the automatic routine in your OR Courseware to apply SUMT to this problem with r _ 103, 102, 10, 1 to find
each of the local maxima. Use x _ 3 and x _ 15 as the initial trial solutions for these searches. Which of the local maxima is a global maximum?