Trial and Error!

Let $x$ be a real number such that the five numbers $\cos(2 \pi x)$, $\cos(4 \pi x)$, $\cos(8 \pi x)$, $\cos(16 \pi x)$, and $\cos(32 \pi x)$ are all nonpositive. What is the smallest possible positive value of $x$?

Source: Math Prize for Girls