EXAMPLE 14 Extra Examples ) Given two positive real numbers x and y. their arithmetic mean is (x + y)/2 and their geometric mean is 4, /xy. When we compare the arithmetic and geometric means of pairs of distinct positive real numbers, we find that the arithmetic mean is always greater than the geometric mean. [For example, whenx=4and y=6, wehave 5=(4+6)/2 > y/4-6 = @.] Can we prove that this inequality is always true? Solution: To prove that (x4 y)/2 > y/xy when x and y are distinct positive real numbers, we can work backward. We construct a sequence of equivalent inequalities. The equivalent inequalities are (@ +y)/2 > \\fx, (x+y)7/4> xy, (x +) > 4y, a2+ 2y + ! > day, -2+ >0, (xy)* > 0| Because (x F)z > 0 when x # y. it follows that the final inequality is true. Because all these inequalities are equivalent, it follows that (x + y)/2 > V/ when x # y. Once we have carried out this backward reasoning, we can build a proof based on reversing the steps. This produces construct a proof using forward reasoning. (Note that the steps of our backward reasoning will not be part of the final proof. These steps serve as our guide for putting this proof together.) Proof: Suppose that x and y are distinct positive real numbers. Then (xv)2 >0 be- cause the square of a nonzero real number is positive (see Appendix 1). Because (x },}1 x% 2xy +y7, this implies that x* 2xy +y* > 0. Adding 4xy to both sides, we obtain x* + 2xy + 3 > 4xy. Because x* + 2xy + ? = (x + y), this means that (x + y)> > 4xy. Dividing both sides of this equation by 4, we see that (x + )>/4 > xy. Finally, taking square roots of both sides (which preserves the inequality because both sides are positive) yields (x +y)/2 > V/ We conclude that if x and y are distinct positive real numbers, then their arithmetic mean (x + y)/2 is greater than their geometric mean V/x_y L Question B4: Look at Example 14 of section 1.8 (A proof of the AM-GM Inequality). One of the hypotheses is that r and y are positive real numbers. In fact, if and y are negative real numbers, then the implication is false. One counterexample is x = 9 and y = 4, for which we have 2% = 6.5