Let's analyze each statement one by one.

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

To determine if $f(x)$ has an absolute maximum on $[-8, 3]$, we can use the Extreme Value Theorem (EVT). The EVT states that if a function is continuous on a closed interval, it will attain both an absolute maximum and minimum within that interval.

1. The function $f(x) = x^4 \cos x + e^{2x^5} - 6$ is continuous on $\mathbb{R}$ because it is composed of continuous functions (polynomial, cosine, and exponential).
2. Since $f(x)$ is continuous on $[-8, 3]$, by the EVT, $f(x)$ will attain an absolute maximum on this interval.

So, statement (i) is true.

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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 $[-4, 6]$.

1. The Extreme Value Theorem requires continuity on a closed interval for a function to guarantee both an absolute maximum and minimum within that interval.
2. If $f(x)$ is not continuous on $[-4, 6]$, then it does not satisfy the EVT's conditions, and there is no guarantee that both an absolute maximum and minimum exist.

Therefore, statement (ii) is true.

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

1. Differentiability on $(-\infty, \infty)$ implies continuity on $(-\infty, \infty)$ but does not necessarily guarantee that an absolute minimum exists on any specific interval.
2. While $f(x)$ is continuous and differentiable on $(-\infty, \infty)$, this fact alone does not ensure an absolute minimum on $[-2, 5]$ without further information about the behavior of $f(x)$ on this interval.
3. Therefore, differentiability on $(-\infty, \infty)$ alone does not imply that an absolute minimum exists on $[-2, 5]$, so statement (iii) is false.

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Conclusion

Based on the analysis above:

• Statement (i) is true.
• Statement (ii) is true.
• Statement (iii) is false.

The correct answer is therefore: (D) (i) and (ii) only.

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Would you like more details or have any questions about this solution?

Related Questions

1. What conditions are required for a function to have an absolute maximum or minimum on an interval?
2. How does the Extreme Value Theorem apply to differentiable functions?
3. Can a function that is differentiable everywhere fail to have an absolute minimum on a closed interval?
4. Why does continuity on an interval play a crucial role in determining maximum and minimum values?
5. How does the behavior of exponential functions impact continuity and extremum conditions?

Tip:

To check if a function has absolute extrema on a closed interval, always ensure it is continuous on that interval.