Consider the following nonconvex programming problem:
Maximize f(x) = 1,000x – 400x2 + 40x3 – x4,
Subject to x2 + x ≤ 500 and 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 bisection method with ϵ = 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?