Calculus – Prove $ – frac {1} {8}


To let $ f (x) $ be convex $ (1, + infty) $ and decrease monotonously to zero. $ f (1) = 1 $ and $ f left ( frac {3} {2} right) = frac {2} {3}. $ To prove $$ – frac {1} {8} < int_1 ^ {+ infty} left (x- lfloor x rfloor – frac {1} {2} right) f (x) , { rm d} x <- frac {1} {18}. $$

Indeed, $ varphi (x): = x- lfloor x rfloor – frac {1} {2} $ is a periodic function with a period $ 1 $. So we have to rethink his behavior $ (1,2) $, which is $$ varphi (x) = x-1 (1 leq x leq 2). $$
That's why $$ varphi (x) = x-n-1 (n leq x leq n + 1) $$
But how to estimate $ f (x) $?