We are tasked with finding the maximum and minimum values of the function $g(\theta) = 6\theta - 8\sin(\theta)$ on the interval $[0, \pi]$.

Steps to solve:

1. Differentiate the function: To find the critical points, we need the derivative of $g(\theta)$. $g'(\theta) = \frac{d}{d\theta} \left( 6\theta - 8\sin(\theta) \right) = 6 - 8\cos(\theta)$

2. Set the derivative equal to zero to find critical points: $6 - 8\cos(\theta) = 0$ $\cos(\theta) = \frac{6}{8} = 0.75$ Solving $\theta = \cos^{-1}(0.75)$, we get: $\theta = \cos^{-1}(0.75) \approx 0.7227$ So, one critical point is $\theta \approx 0.7227$.

3. Evaluate the function at the endpoints $\theta = 0$ and $\theta = \pi$, and at the critical point $\theta = 0.7227$:

   • At $\theta = 0$: $g(0) = 6(0) - 8\sin(0) = 0$
   • At $\theta = \pi$: $g(\pi) = 6\pi - 8\sin(\pi) = 6\pi \approx 18.8496$
   • At $\theta = 0.7227$: $g(0.7227) = 6(0.7227) - 8\sin(0.7227)$ Using a calculator: $g(0.7227) \approx 4.336 - 8(0.6614) \approx 4.336 - 5.291 = -0.955$

4. Determine the maximum and minimum values:

   • Minimum value: $g(0.7227) \approx -0.955$
   • Maximum value: $g(\pi) \approx 18.8496$

Thus, the minimum value is approximately -0.955, and the maximum value is approximately 18.8496.

Would you like more details or have any questions?

Related questions:

1. What are critical points and why do we use them to find extreme values?
2. How is the derivative used to identify increasing and decreasing behavior of a function?
3. Why is it necessary to evaluate the function at the endpoints of the interval?
4. Can we solve this problem graphically to verify our result?
5. How would the solution change if the interval was different, for example, $[0, 2\pi]$?

Tip:

When solving optimization problems on a closed interval, always check the function at the endpoints as well as at any critical points.