# 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 $$f (x)$$?