Problem 3 (20 points). Part I (9 points) Find ALL local and global optimal solutions of the following minimization problems, whose objective function is denoted by f and feasible set by X (do NOT use Matlab or any other solver): (a). f(x) = sin x and X = [0, 3π]; (b). f(x) = sin x and X = [0, 4π]; (c). f(x) = sin x and X = [−π/4,(11/4)π]; (d). f(x) = max(0, x2 − 1) and X = (−[infinity], [infinity]); (e). f(x) = ( 1 x · sin (x), if x 6= 0 0, if x = 0; and X = [0,(2/3)π];

You drop a rubber ball off the roof of a 50 meter high building onto a paved parking lot. it bounces back up with every bounce, but not quite all the way back up to you. after the first bounce it bounces back only 80 percent of the distance it was dropped from. the pattern continues, meaning after every bounce it comes up to just 80 percent of the previous maximum height. so if before the first bounce the height is 50 meters, what height does the ball reach after the fifth bounce? round your answer to one decimal place and chose the correct response from the choices below:

Factor the polynomial 4x4 – 20x2 – 3x2 + 15 by grouping. what is the resulting expression? (4x2 + 3)(x2 – 5) (4x2 – 3)(x2 – 5) (4x2 – 5)(x2 + 3) (4x2 + 5)(x2 – 3)

What is the greatest number of parts of a circle that can be formed by cutting the circle with 7 straight cuts? (note: the parts do not have to be equal in size)