Answer
The graph of $y=2x+\sin x$ does not have any horizontal tangents on $[0,2\pi]$.
Work Step by Step
$$y=f(x)=2x+\sin x$$
To know whether the graph of $f(x)$ has any horizontal tangents on $[0,2\pi]$ or not, we rely on the fact that horizontal tangents are the only ones which possess the slope value $0$. So, by taking the derivative of $f(x)$, we will see whether the derivative can obtain the value $0$ or not on $[0,2\pi]$.
$$f'(x)=(2x+\sin x)'=2+\cos x$$
We have $f'(x)=0$ when $$2+\cos x=0$$ $$\cos x=-2$$
The value of $\cos x$ is limited to the range $[-1,1]$. In other words, there is no value of $x$ for which $\cos x=-2$.
So the graph of $f(x)$ does not have any horizontal tangents on $[0,2\pi]$.
