What values of x make the graph of f(x) have a horizontal tangent?

[tmt1]

For what values of does the graph of have a horizontal
tangent?
f(x) = x + 2sin(x)

I get this:

$f'(x) = 1 + 2 \cos(x)$

And I understand that I need to set this to zero.

$1 + 2cosx = 0$

$cosx = 1/2$

How do I isolate x in this situation?

 

[Taschee]

First of all, the equations is cos(x) = -1/2;

arccos gives you cos(\pi/3) = 0.5;

If you draw the graph of cos(x), this might help find you all the values for x where cos(x) = -0.5

 

[MarkFL]

Yes, as mentioned, you wind up with:

\(\displaystyle \cos(x)=-\frac{1}{2}\)

Now, there is a quadrant II and a quadrant III solution, given generally by:

\(\displaystyle x=(2k+1)\pi\pm\frac{\pi}{3}=\frac{\pi}{3}\left(3(2k+1)\pm1\right)\) where \(\displaystyle k\in\mathbb{Z}\)