Let's analyze each of the statements provided in Problem #3 to determine which are always true.

1. 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 is always true, we note that the function $f(x)$ is continuous on the closed interval $[-8, 3]$. By the Extreme Value Theorem, a continuous function on a closed interval will attain both absolute maximum and minimum values within that interval. Since $f(x)$ is continuous on $[-8,3]$, it will indeed have an absolute maximum. Therefore, statement (i) is true.

2. 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 $[-4,6]$.

   • If $f(x)$ is not continuous on $[-4,6]$, then the Extreme Value Theorem does not guarantee the existence of absolute extrema. However, it's possible for a non-continuous function to still have an absolute maximum and minimum, depending on the nature of the discontinuities. Therefore, statement (ii) is not always true, as we cannot make a definitive statement without additional information about the function's discontinuities.

3. 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 $(-\infty, \infty)$ and particularly on $[-2,5]$. By the Extreme Value Theorem, since $f(x)$ is continuous on the closed interval $[-2,5]$, it must attain both an absolute minimum and maximum on that interval. Thus, statement (iii) is true.

Conclusion

The statements that are always true are (i) and (iii).

Answer

The correct answer is (B) (i) and (iii) only.

Would you like a detailed explanation of each step or have any questions?

Here are some related questions to deepen your understanding:

1. What is the Extreme Value Theorem, and why is it essential in this problem?
2. Can a function with a removable discontinuity still have absolute extrema on a closed interval?
3. How does differentiability imply continuity, and why does it matter in the context of extrema?
4. Why doesn’t continuity guarantee absolute extrema if the interval is open rather than closed?
5. What are some examples of functions that have absolute maximums or minimums despite being non-continuous?

Tip: Always check if the interval is closed and if the function is continuous on it when determining absolute extrema.