This question involves analyzing three functions and determining which have a maximum value of 6. Let's go through each of the given functions:

1. $f(x) = - \cos(x) + 7$
   The cosine function ($\cos(x)$) oscillates between -1 and 1. When we apply a negative sign in front, the range becomes between -1 and 1, but inverted. Adding 7 shifts this range by 7 units. So, the function $f(x)$ oscillates between $6$ and $8$, with a maximum value of 7.

2. $g(x) = 2 \sin(x) + 4$
   The sine function ($\sin(x)$) oscillates between -1 and 1. Multiplying it by 2 changes the range to between -2 and 2. Adding 4 shifts this range to between 2 and 6. Therefore, the maximum value of $g(x)$ is 6.

3. $h(x) = -8 \cos(x) - 2$
   The cosine function ($\cos(x)$) oscillates between -1 and 1. Multiplying it by -8 gives a range between -8 and 8, but inverted. Adding -2 shifts the range to between -10 and 6. Therefore, the maximum value of $h(x)$ is 6.

Conclusion:

The functions that have a maximum value of 6 are g(x) and h(x).

So the correct answer is D: II and III only.

Let me know if you need further clarification or details!

Follow-up questions:

1. How does shifting a trigonometric function affect its maximum or minimum values?
2. What happens to the range of a function when you multiply by a constant?
3. What is the general form of a sinusoidal function with a maximum value?
4. How would you adjust a sine or cosine function to achieve a maximum value of 10?
5. Can these functions also have a minimum value of 6? How would the transformations change?

Tip: When analyzing trigonometric functions, always pay attention to how the transformations (such as scaling or shifting) affect the range!