Determine the points of inflection of the function.

f(x) = x + sin x (−2π ≤ x ≤ 2π)

I'm still completely confused

Determining the points of inflection of a function involves finding the x-values where the concavity changes. In simple terms, it's where the function changes from being concave up (like a U-shape) to concave down (like an upside-down U-shape), or vice versa.

To find the points of inflection of the function f(x) = x + sin(x) over the given interval, you need to follow these steps:

Step 1: Find the first derivative of the function f(x).

The first derivative of f(x) will tell you where the function is increasing or decreasing. So, let's differentiate f(x):

f'(x) = 1 + cos(x)

Step 2: Find the second derivative of the function f(x).

The second derivative of f(x) will give you information about the concavity of the function. So, differentiate f'(x):

f''(x) = -sin(x)

Step 3: Set the second derivative equal to zero and solve for x.

To find the points of inflection, we need to find the x-values where the second derivative is equal to zero.

- sin(x) = 0

The sine function is zero at x = 0, π, and 2π. These are the critical points of the function.

However, we also need to check for any x-values where the second derivative does not exist or is undefined. In this case, sin(x) is well-defined for all x-values, so there are no such x-values.

Step 4: Analyze the concavity around the critical points.

To determine the concavity around the critical points, evaluate the second derivative for values less than and greater than each critical point:

For x < 0: f''(x) < 0 (since sin(x) is negative in this interval)
For 0 < x < π: f''(x) > 0 (since sin(x) is positive in this interval)
For π < x < 2π: f''(x) < 0 (since sin(x) is negative in this interval)
For x > 2π: f''(x) > 0 (since sin(x) is positive in this interval)

Step 5: Determine the points of inflection.

Based on the analysis of the concavity, the points of inflection occur at x = 0, π, and 2π.

So, the points of inflection for the given function f(x) = x + sin(x) over the interval -2π ≤ x ≤ 2π are x = 0, π, and 2π.