Consider a question and two statements:<br>Question: Is \frac{x^6+y^6}{x^4+y^4} <strong>ALWAYS</strong> greater than \frac{x^4+y^4}{x^2+y^2} (x \neq y \neq 0)?<br>Statement-I: x \gt y<br>Statement-II: x^2+y^2 \gt 2xy<br>Which one of the following is correct in respect of the question and the statements?
- A. Statement-I alone is required to answer the question
- B. Statement-II alone is required to answer the question
- C. Both Statement-I and Statement-II are required to answer the question
- D. Neither Statement-I nor Statement-II is required to answer the question ✓
Correct Answer: D. Neither Statement-I nor Statement-II is required to answer the question
Explanation
Let a=x^2 and b=y^2. We check if \frac{a^3+b^3}{a^2+b^2} \gt \frac{a^2+b^2}{a+b}. Cross multiplying gives (a^3+b^3)(a+b) \gt (a^2+b^2)^2. This simplifies to a^4+ab^3+a^3b+b^4 \gt a^4+2a^2b^2+b^4, or ab(a-b)^2 \gt 0. Since squares a and b are positive, this is inherently true. Neither statement is required.
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