Problem: a β©Ύ 1 a β©Ύ 1
a β a + x = x a β a + x β β = x
is equal to
Answer Choices:
A. a β 1 a β β 1
B. a β 1 2 2 a β β 1 β
C. a β 1 a β 1 β
D. a β 1 2 2 a β 1 β β
E. 4 a β 3 β 1 2 2 4 a β 3 β β 1 β
Solution:
Since x x x β©Ύ 0 x β©Ύ 0 x β©Ύ 0 x β©Ύ 0
a β a + x = x 2 a β a + x β = x 2
Since the left member of this equation is a decreasing function of x x y = a + x y = a + x β
a β y = x 2 a β y β y 2 = x 2 β y 2 a β y β ( a + x ) = x 2 β y 2 β ( x + y ) = ( x + y ) ( x β y ) 0 = ( x + y ) ( x β y + 1 ) a β y = x 2 a β y β y 2 = x 2 β y 2 a β y β ( a + x ) = x 2 β y 2 β ( x + y ) = ( x + y ) ( x β y ) 0 = ( x + y ) ( x β y + 1 ) β
Since a β©Ύ 1 a β©Ύ 1 x β©Ύ 0 x β©Ύ 0 y > 0 y > 0 x + y β 0 x + y ξ = 0
x β y + 1 = 0 x + 1 = y x + 1 = a + x ( x + 1 ) 2 = a + x x = β 1 Β± 4 a β 3 2 . x β y + 1 = 0 x + 1 = y x + 1 = a + x β ( x + 1 ) 2 = a + x x = 2 β 1 Β± 4 a β 3 β β . β
The positive solution x = 4 a β 3 β 1 2 x = 2 4 a β 3 β β 1 β