Let's solve the problem step by step.

Part a: Find the local extrema of the function

The function given is:

$f(x) = \cos(3x) + \sqrt{3} \sin(3x)$

We are tasked with finding the local extrema on the interval $0 \leq x \leq \frac{2\pi}{3}$.

Step 1: Take the derivative of $f(x)$

We first find the derivative of the function to locate the critical points. Using the chain rule and the fact that the derivative of $\cos(3x)$ is $-3\sin(3x)$ and the derivative of $\sin(3x)$ is $3\cos(3x)$, we get:

$f'(x) = -3\sin(3x) + 3\sqrt{3}\cos(3x)$

Step 2: Set $f'(x) = 0$ to find critical points

To find the critical points, set the derivative equal to zero:

$-3\sin(3x) + 3\sqrt{3}\cos(3x) = 0$

Dividing through by 3 simplifies this to:

$-\sin(3x) + \sqrt{3}\cos(3x) = 0$

Rearranging gives:

$\sin(3x) = \sqrt{3}\cos(3x)$

Dividing both sides by $\cos(3x)$ (assuming $\cos(3x) \neq 0$):

$\tan(3x) = \sqrt{3}$

This gives:

$3x = \tan^{-1}(\sqrt{3})$

We know from trigonometry that $\tan^{-1}(\sqrt{3}) = \frac{\pi}{3}$, so:

$3x = \frac{\pi}{3} \quad \Rightarrow \quad x = \frac{\pi}{9}$

Step 3: Check the interval

The critical point $x = \frac{\pi}{9}$ lies within the interval $0 \leq x \leq \frac{2\pi}{3}$.

To determine whether this point is a local maximum or minimum, we can use the second derivative test or analyze the sign of $f'(x)$ around $x = \frac{\pi}{9}$.

Step 4: Endpoints

The endpoints of the interval are $x = 0$ and $x = \frac{2\pi}{3}$. We should check the function values at these points to find any global extrema within the interval.

Part b: Graph the function and its derivative

1. The function is periodic, so plotting $f(x) = \cos(3x) + \sqrt{3}\sin(3x)$ and its derivative $f'(x)$ on the interval $[0, \frac{2\pi}{3}]$ will show how the function behaves, specifically how the local extrema relate to the derivative.

2. The derivative $f'(x)$ changes sign at critical points, which indicates changes from increasing to decreasing or vice versa. By graphing both $f(x)$ and $f'(x)$, we can comment on how the sign of $f'(x)$ corresponds to the increasing or decreasing nature of $f(x)$.

Would you like a detailed graph for part b? Let me know if you need further clarifications, or if you'd like to proceed with a graph!