How to find value(s) for $~x~$ where $~y~$ has a horizontal tangent line if $$y = \frac{1}{2}x+\text{sin }x~?$$
I understand how to find the equation of a horizontal tangent line, but I'm not even sure where to start with this. All help is appreciated. The question also specifies to use radians.
A horizontal line has a slope of zero and if it’s tangent to our curve, our derivative has to be zero: $$y’(x)=0.5+\cos(x)\quad \Rightarrow\quad 0=0.5+\cos(x)$$ This yields $x=\frac{2\pi}{3}$ and other terminal angles ($+2\pi k$ as $k\in\mathbb{Z}$). For example, $x=-\frac{4\pi}{3}, x=\frac{8\pi}{3}$ etc.