MHB JustWar's question at Yahoo Answers regarding horizontal tangents

[MarkFL]

Here is the question:

Horizontal tangents?

For what values of x in [0,2π] does the graph of f(x)=x+2sinx have a horizontal tangent?
List the values of x below. Separate multiple values with commas.

x=

I have posted a link there to this thread so the OP can view my work.

 

[MarkFL]

Hello JustWar,

We are given the function:

$$f(x)=x+2\sin(x)$$

To find where there are horizontal tangents, we must equate the first derivative to zero, and then solve for $x$ over the given interval.

$$f'(x)=1+2\cos(x)=0\,\therefore\,\cos(x)=-\frac{1}{2}$$

Hence, we find:

$$x=\frac{2\pi}{3},\,\frac{4\pi}{3}$$

Here is a plot of $f(x)$ over the given interval and the resulting tangent lines:

$$y_1=\frac{2\pi}{3}+\sqrt{3}$$

$$y_2=\frac{4\pi}{3}-\sqrt{3}$$

View attachment 2206