The best way to answer these kind of questions is to insert numbers and verify whether the inequalities hold true.

Take x = 100, y = 5, z = 2,the values satisfy 100 > 5^2 > 2^4 and the inequality x > y > z, but the other two inequalities don`t hold. So eliminate II and III options.
Take x = 0.1, y = 0.2, z = 0.3, the values satisfy 0.1 > 0.2² > 0.3^4. But these values does not satisfy x > y > z. So eliminate I option

The correct answer is none of the above.

Anurag@Gurome wrote:The best way to answer these kind of questions is to insert numbers and verify whether the inequalities hold true.

(I) Take x = 100, y = 5, z = 2, the inequality x > y > z holds true.
(II) Take x = 0.1, y = 0.2, z = 0.3, it satisfies 0.1 > 0.2² > 0.3^4. So, z > y > x also holds true.
(III) Take x = 1, y = 0.2 and z = 0.3, it satisfies 1 > 0.2² > 0.3^4. So, x > z > y also holds true.

The correct answer is I, II, and III.

But the question asked "which of the following must be true". Clearly from your examples, none of the three options absolutely must be true, so the answer is "none of the above".

If the question asks "which of the following could be true", then the answer is I, II and III, as your examples prove.

I agree with Ian and edited my reply.
Thanks Ian for pointing it.