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It appears you have uploaded a photograph with several math problems. Let's address each question one by one, starting with problem number 23 related to the function \( f(x) = x - 2\sin x \) over the interval \( 0 \leq x \leq 3\pi \).

### Part (a): Find the interval on which \( f \) is increasing or decreasing.

To determine where the function is increasing or decreasing, we first need to find its derivative and analyze the sign of the derivative:

\[ f'(x) = 1 - 2\cos x \]

We'll find the values of \( x \) for which \( f'(x) = 0 \) to identify critical points:

\[ 1 - 2\cos x = 0 \]
\[ \cos x = \frac{1}{2} \]

The cosine function equals \( \frac{1}{2} \) at specific standard angles within the given interval \( 0 \leq x \leq 3\pi \):

\[ x = \frac{\pi}{3}, \frac{5\pi}{3}, \frac{7\pi}{3}, \frac{11\pi}{3} \]

Next, we analyze the sign of \( f'(x) \) in the intervals defined by these critical points to determine where the function is increasing (where \( f'(x) > 0 \)) and decreasing (where \( f'(x) < 0 \)).

### Part (b): Find the local maximum and minimum values of \( f \).

The local maxima and minima occur at these critical points where the derivative changes sign. We can calculate \( f(x) \) at these points and check the sign changes in \( f'(x) \) around them to identify maxima and minima.

I'll now perform these calculations to complete both parts (a) and (b) of problem 23.

Given f(x) = x - 2sin(x), 0 ≤ x ≤ 3π, find the interval on which f is increasing or decreasing and find the local maximum and minimum values of f.

Analyze the function f(x) = x - 2sin(x) over the interval 0 ≤ x ≤ 3π. Learn how to find the intervals where the function is increasing or decreasing and determine the local maximum and minimum values using derivative-based techniques.

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