Consider the function on the interval (0, 2π). f(x) = x − 2 sin x (a) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.)

Increasing:

Decreasing

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Step-by-step explanation:

Increasing and Decreasing Intervals

To find if a function is increasing in a point x=a, we evaluate the first derivative in x=a and if:

For continuous functions, we can safely assume between a given critical point and the next one, the function keeps its behavior, i.e. it's increasing or decreasing in the interval formed by both points.

So, we find the critital points of  

Taking the derivative

Equating to 0

Solving

There are two solutions in the interval  

Now we compute the second derivative

Evaluating for both critical points

Since it's positive, the point is a minimum

Since it's negative, the point is a maximum

In the interval  

the function is decreasing

In the interval  

the function is increasing

In the interval  

the function is decreasing

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