Answer
The graph of $y=x+\sin x$ has one horizontal tangent at $x=\pi$ on $[0,2\pi]$.
Work Step by Step
$$y=f(x)=x+\sin x$$
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 $[0,2\pi]$.
$$f'(x)=(x+\sin x)'=1+\cos x$$
We have $f'(x)=0$ when $$1+\cos x=0$$ $$\cos x=-1$$
On $[0,2\pi]$, there is the value of $x=\pi$ whose $\cos x=-1$.
So the graph of $f(x)$ has one horizontal tangent at $x=\pi$ on $[0,2\pi]$.
