A problem with an elementary function

Question

Prove or disprove:

For every $n\in\mathbb N$ there exist $x_n\in (10^n,+\infty)$ such that $f(x_n)<10^{-n}$ where $f(x)=(x \sin x - \cos^2 x)^2 x^6 + x^2 \cos^8 x.$

Answer 1

We answer the question somewhat reluctantly, since there is undoubtedly a typo.

Let $n=3$, for no good reason. We show that there is no $x_n \gt 1000$ at which our function is $\lt 1/1000$.

For note from $\cos^2 x+\sin^2 x=1$ that one of $\cos^2 x$ or $\sin^2 x$ is $\ge 1/2$.

If $\cos^2 x\ge 1/2$, then our expression is greater than $(1000)^2(1/2)^4$. This is because the second term is greater than $(1000)^2(1/2)^4$, and the first term is non-negative.

If $\sin^2 x \ge 1/2$, then our expression is $\ge (500\sqrt{2}-1)^2(1000)^6$.

So for all $x\gt 1000$, our function is quite large, much larger than $1/1000$.