The problem asks to prove that $\mathbb{E}[X^n] \geq (\mathbb{E}[X])^n$ for $n = 2k$, and it provides multiple options to determine which argument proves the claim.

Breakdown of the options:

1. Option (a): Uses Jensen's inequality.

   • Jensen's inequality states that for a convex function $f$, $f(\mathbb{E}[X]) \leq \mathbb{E}[f(X)]$. In this case, it is possible to apply Jensen's inequality to show that $\mathbb{E}[X^n] \geq (\mathbb{E}[X])^n$, as the function $x^n$ (for even $n$) is convex. This suggests that this argument is valid.

2. Option (b): Refers to the Mean Value Theorem.

   • The Mean Value Theorem typically applies to differentiable functions, but its direct application here seems unlikely. This option is not commonly associated with inequalities like the one in the problem.

3. Option (c): Provides a comparison using $\mathbb{E}[X^n]^{1/n}$.

   • This approach involves taking the $n$-th root and relies on properties of expectations. However, while $\mathbb{E}[X^n]^{1/n} \geq \mathbb{E}[X]$ can be valid, it doesn’t directly prove the specific inequality for even powers $n = 2k$.

4. Option (d): Compares the variance $\text{Var}(X^k)$.

   • This argument compares variances and uses an inequality involving variances, which can show $\mathbb{E}[X^n] \geq (\mathbb{E}[X])^n$ under certain conditions. This might also be a valid approach, depending on how it is structured.

5. Option (e): Suggests that none of these prove the claim.

   • This could be the correct answer if none of the arguments work fully.

Conclusion:

Option (a) appears to be the most appropriate due to the use of Jensen’s inequality, which is a common tool for proving inequalities involving expectations of convex functions. Therefore, (a) seems to provide the correct reasoning.

Would you like further details or have any questions about Jensen's inequality or the other methods mentioned? Here are some related questions you might explore:

1. How does Jensen's inequality apply to different convex functions?
2. Can you explain why the Mean Value Theorem isn't useful in this case?
3. How can we derive inequalities for random variables using variance comparisons?
4. Is there a way to generalize the result $\mathbb{E}[X^n] \geq (\mathbb{E}[X])^n$ for odd $n$?
5. What are other common inequalities used in probability theory to compare expectations?

Tip: When working with inequalities in probability, always check whether the function you're dealing with is convex or concave, as this can give you immediate tools like Jensen's inequality to apply!