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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!

Find the local extrema of the function f(x) = cos(3x) + √3 sin(3x) on the interval 0 ≤ x ≤ 2π/3, and say where they occur. Graph the function and its derivative together. Comment on the behavior of f in relation to the signs and values of f'.

Solve for the local extrema of the function f(x) = cos(3x) + √3 sin(3x) within the interval [0, 2π/3]. This calculus problem involves finding critical points by setting the derivative equal to zero and applying trigonometric identities like tan(3x) = √3. A detailed graph of the function and its derivative is discussed.

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