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.