To find the local minima and maxima of the function $f(x)$, given its derivative $f'(x) = 30 \sin(3x)$ over the interval $\left[ -\frac{4\pi}{3}, \frac{4\pi}{3} \right]$, we follow these steps:

1. Critical Points (where $f'(x) = 0$):

To find the critical points, set $f'(x) = 0$:

$30 \sin(3x) = 0$

This simplifies to:

$\sin(3x) = 0$

The sine function equals zero at integer multiples of $\pi$, so:

$3x = n\pi \quad \text{where} \, n \in \mathbb{Z}$

Solving for $x$:

$x = \frac{n\pi}{3}$

Now, we need to find the values of $n$ such that $x$ lies within the interval $\left[ -\frac{4\pi}{3}, \frac{4\pi}{3} \right]$.

For $-\frac{4\pi}{3} \leq \frac{n\pi}{3} \leq \frac{4\pi}{3}$:

Multiplying through by 3:

$-4\pi \leq n\pi \leq 4\pi$

Dividing by $\pi$:

$-4 \leq n \leq 4$

Thus, $n = -4, -3, -2, -1, 0, 1, 2, 3, 4$.

Now, we substitute these values of $n$ into $x = \frac{n\pi}{3}$ to get the critical points:

$x = \left\{ -\frac{4\pi}{3}, -\pi, -\frac{2\pi}{3}, -\frac{\pi}{3}, 0, \frac{\pi}{3}, \frac{2\pi}{3}, \pi, \frac{4\pi}{3} \right\}$

2. Sign of $f'(x)$ to Determine Increasing/Decreasing Intervals:

To determine whether the function is increasing or decreasing, analyze the sign of $f'(x) = 30 \sin(3x)$ in each interval between the critical points. The sine function oscillates between $-1$ and $1$, so:

• $f'(x) > 0$ when $\sin(3x) > 0$ (function is increasing).
• $f'(x) < 0$ when $\sin(3x) < 0$ (function is decreasing).

We know that the sine function has a period of $\frac{2\pi}{3}$, and alternates between positive and negative values in successive intervals.

3. Local Minima and Maxima:

To identify the nature of each critical point (local minimum or maximum), examine the sign changes of $f'(x)$:

• If $f'(x)$ changes from positive to negative at a critical point, it's a local maximum.
• If $f'(x)$ changes from negative to positive at a critical point, it's a local minimum.

4. Summary of Local Minima/Maxima:

Without graphing each interval explicitly, we can say that:

• Local minima occur where $f'(x)$ changes from negative to positive.
• Local maxima occur where $f'(x)$ changes from positive to negative.

5. Local Minima:

From the sine wave's behavior and the critical points found, the local minima occur at:

$x = -\frac{2\pi}{3}, 0, \frac{2\pi}{3}$

Thus, the local minima are at $x = -\frac{2\pi}{3}, 0, \frac{2\pi}{3}$.

Would you like more details, or do you have any questions?

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Here are 5 related questions for further exploration:

1. How do you find the second derivative to confirm concavity at the critical points?
2. How does the period of the sine function affect the intervals of increase and decrease?
3. What is the general behavior of a function whose derivative is a sine function?
4. How can you determine the global maximum or minimum if the function is restricted to a closed interval?
5. How do the local minima and maxima compare to the graph of the sine function?

Tip: When solving problems involving trigonometric derivatives, always consider the period and symmetry of the trigonometric function to understand the intervals of increase and decrease effectively.