Problems for Section 10.2

1. Let a and b be real numbers. Prove that  and that  if and only if
a = b = 0.

2. Let c₁, c₂, ..., c_n be real numbers. Prove that and  if and only if each c_i = 0.

3. Let f(x) = ax² + bx + c, where a, b, and c are real numbers and a > 0. Let D be the discriminant b² - 4ac. Show the following:

(a) If D < 0, then f(x) > 0 for all x, and f(x) = 0 has no real roots.

(b) If D = 0, then  for all x, and f(x) = 0 has one real root.

(c) If D > 0, then f(x) < 0 for  f(x) > 0 for  or  and f(x) = 0 has two real roots.

4. Let f(x) = ax² + bx + c, where a, b, and c are real numbers and a < 0. Let D = b² - 4ac. Show the following:

(a) If D < 0, then f(x) < 0 for all x, and f(x) = 0 has no real roots.

(b) If D = 0, then  for all x, and f(x) = 0 has one real root.

(c) If D > 0, then f(x) > 0 for  f(x) < 0 for  or  and f(x) = 0 has two real roots.

^*5. Let F₁, F₂, F₃, ... be the sequence of Fibonacci numbers 1, 1, 2, 3, 5, ... and let R_n = F_n+1/F_n for n = 1, 2 , 3, ... . Do the following:

(a) Show that 

(b) Prove that R_2n-1 < R_2n+1 < R_2n and R_2n+1 < R_2n+2 < R_2n for all positive integers n, that is, that R₁ < R₃ < R₅ < R₇ < ... < R₈ < R₆ < R₄ < R₂.

Monday, June 22, 1998