Given that x is a real number, and:
{x}
|x-1| - 1 = ---------
|x-1|
where, [x] is the greatest integer less than or equal to x, and {x}= x-[x]
Find the largest possible value of x.
*** |x| denotes the
absolute value of x.
Let x = n + f, where n is an integer and 0 <= f <1
First, let's hope that the maximum x is >= 1
If so, then (n+f-2) = f/(n+f-1)
Multiplying by (n+f-1) and collecting terms gives:
n^2 + n(2f-3) + (f^2-4f+2) = 0
Solving for n has two roots, the larger of which is
n = (3 - 2f + sqrt(1-8f))/2
This is maximized when f = 0, and fortunately this makes n = 2, which is an integer. Since this is greater than 1, there are no other cases that need to be considered
Hope I didn't make any math mistakes.
Final answer: x = 2
Checking:
a) |2-1| - 1 = 0/|2-1|, so that works!
b) If x is between 2 and 3, say 2 + f, then by substitution we get
f = f/(1+f), which implies f is 0
c) If x is between 3 and 4, say 3 + f, then by substitution we get
1+f = f/(2+f)
But LHS >= 1 and RHS <1
And this only gets worse as X gets bigger.
So x = 2 checks out.
Edited on February 20, 2014, 11:44 pm
Quadratic equation (spoiler)
