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Let’s analyze each statement carefully to determine which ones are always true.

### Statement Analysis

**Statement (i):**  
\( f(x) = x^4 \cos x + e^{2x^5} - 6 \) has an absolute maximum on the interval \([-8, 3]\).

- To determine if this statement is true, we need to consider whether \( f(x) \) is continuous on the closed interval \([-8, 3]\).
- Since \( x^4 \cos x \) and \( e^{2x^5} - 6 \) are continuous functions for all \( x \), their sum, \( f(x) \), is also continuous on \([-8, 3]\).
- By the Extreme Value Theorem, a continuous function on a closed interval must attain both an absolute maximum and an absolute minimum on that interval.
- Therefore, statement (i) is true.

**Statement (ii):**  
If \( f(x) \) is not continuous on \([-4, 6]\), then \( f(x) \) does not have both an absolute maximum and an absolute minimum on \([-4, 6]\).

- For a function to guarantee an absolute maximum and minimum on a closed interval, it must be continuous on that interval.
- If \( f(x) \) is not continuous on \([-4, 6]\), it may not achieve an absolute maximum or minimum because the function could approach an undefined or divergent point within the interval.
- This statement is consistent with the Extreme Value Theorem, which only applies to continuous functions on closed intervals. Thus, if continuity is missing, the function may not attain both an absolute maximum and minimum.
- Therefore, statement (ii) is true.

**Statement (iii):**  
If \( f(x) \) is differentiable on \((-\infty, \infty)\), then \( f(x) \) has an absolute minimum on \([-2, 5]\).

- Differentiability on \((-\infty, \infty)\) implies continuity on \((-\infty, \infty)\), but it does not guarantee that the function will have an absolute minimum or maximum on any specific interval.
- For instance, a function that grows without bound in certain directions (like certain polynomial functions) may not attain an absolute minimum on a closed interval, even if it's differentiable everywhere.
- Therefore, differentiability alone does not ensure the existence of an absolute minimum on a specified interval.
- This makes statement (iii) false.

### Conclusion

The statements that are always true are:
- **(i)** and **(ii)**.

Thus, the correct answer is **(D) (i) and (ii) only**.

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Would you like further clarification on any of the statements or the reasoning?

### Related Questions
1. What is the Extreme Value Theorem, and why is it important for determining maximum and minimum values?
2. How does continuity differ from differentiability, and how does each property affect function behavior?
3. Can a function be differentiable without being continuous? Why or why not?
4. What are the conditions for a function to have both maximum and minimum values on an interval?
5. How does the behavior of exponential and trigonometric functions affect their continuity?

### Tip
For a function to guarantee both an absolute maximum and minimum on a closed interval, it must be continuous on that interval.

Which of the following statements are always true?
(i) f(x) = x^4 cos x + e^(2x^5) - 6 has an absolute maximum on the interval [-8, 3]
(ii) If f(x) is not continuous on [-4, 6], then f(x) does not have both an absolute maximum and an absolute minimum on [-4, 6].
(iii) If f(x) is differentiable on (-∞, ∞), then f(x) has an absolute minimum on [-2,5].

Explore which statements about a function’s absolute maximum, minimum, continuity, and differentiability are always true. Analyze each statement using the Extreme Value Theorem and understand the conditions for absolute extremum on closed intervals. Suitable for college calculus students.

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