How to show that $\frac{3}{2}x-6-\frac{1}{2}\sin(2x)$ has one unique real root?

Question

I want to show that the function $$\frac{3}{2}x-6-\frac{1}{2}\sin(2x)$$ has one unique real root.

By looking at the graph, it is obvious that there is only one root. But how can I show that this is unique real root by a formal mathematical proof?

Answer 1

Use the graph to find points of opposite sign for the intermediate value theorem.

Use the derivative to prove strict monotonicity.

Answer 2

Rolle's theorem is enough. It is quite trivial that a root must exist by continuity, since $3x-12$ is unbounded while $\sin(2x)\in[-1,1]$. On the other hand, by assuming that $3x-12-\sin(2x)$ has two roots $a,b$, then the derivative $3-2\cos(2x)$ must vanish for some $x\in(a,b)$. That cannot happen, since $\cos(2x)\in[-1,1]$, too.