Consider the following in respect of a positive real number x : I. x + \frac{1}{x} > 1 II. x^2 + \frac{1}{x^2} > 2 III. \left(x + \frac{1}{x}\right)^4 > 9 Which of the above are correct?
- A. I and II only
- B. II and III only
- C. I and III only
- D. I, II and III ✓
Correct Answer: D. I, II and III
Explanation
For any positive real number x, A.M \ge G.M implies x + \frac{1}{x} \ge 2. Thus, x + \frac{1}{x} > 1 is true. Also x^2 + \frac{1}{x^2} \ge 2 is true. And (x + \frac{1}{x})^4 \ge 2^4 = 16 > 9, which is also true. Thus all statements are correct.
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