MHB JustWar's question at Yahoo Answers regarding horizontal tangents

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The function f(x) = x + 2sin(x) has horizontal tangents where its first derivative, f'(x) = 1 + 2cos(x), equals zero. This occurs when cos(x) = -1/2, leading to the solutions x = 2π/3 and x = 4π/3 within the interval [0, 2π]. A plot of the function illustrates the tangent lines at these points. The values of x where the graph has horizontal tangents are therefore 2π/3 and 4π/3. Understanding these points is crucial for analyzing the behavior of the function.
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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.
 
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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
 

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