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If x > y^2 > z^4, which of the folloing statements could be true? I. x > y > z II. z > y > x III. x > z > y
A. I only B. I and II only C. I and III only D. II and III only E. I, II, and III
Yes, just plugging in numbers work best for such questions. The only thing to keep in mind is that you should plug in the right numbers. How do you know the right numbers? When I see x > y^2 > z^4, I think that y^2 and z^4 are non negative. Since y^2 > z^4, y^2 cannot be 0. Only z can be 0. x has to be positive. Also, I have to take into account two ranges: 0 to 1 and 1 to infinity. The powers behave differently in these two ranges. I will consider negative numbers only if I have to since with powers, they get confusing to deal with.
The question says: "Which of the following could be true?" We have to find examples where each relation holds.
I. x > y > z This is the most intuitive of course. z = 0, y = 1 and x = 2 2 > 1^2 > 0^4
II. z > y > x Let me consider the 0 to 1 range here. Say z = 1/2, y = 1/3 and x = 1/4 1/4 > 1/9 > 1/16
III. x > z > y Let's stick to 0 to 1 range. z > y as in case II above but x has to be greater than both of them. Say z = 1/2, y = 1/3 and x = 1 1>1/9 > 1/16