This exercise completes a detail mentioned in Example 4.

Chapter 4, Problem 43

(choose chapter or problem)

This exercise completes a detail mentioned in Example 4. In that example we used the result that the input minimizing the function D(x) is the same input that minimizes the function y x2 x 1. For this result to be applicable, we need to know that the quantity x2 x 1 is nonnegative for all x. Parts (a) and (b) each suggest a way to verify that this is indeed the case. [Actually, parts (a) and (b) both show that the quantity x2 x 1 is positive for all x.] (a) Use the fact that x2 x 1 can be written as (x ) 2 to explain (in complete sentences) why the quantity x2 x 1 is positive for all inputs x. (b) Compute the discriminant of the quadratic x2 x 1 and explain why the graph of y x2 x 1 has no x-intercepts. Then use this fact to explain why the quantity x2 x 1 is always positive. Hint: Is the graph of the parabola y x2 x 1 U-shaped up or U-shaped down? 44. Let f(x)