Consider the following inequalities : 1. \frac{a^{2}-b^{2}}{a^{2}+b^{2}} \gt \frac{a-b}{a+b} 2. \frac{a^{3}+b^{3}}{a^{2}+b^{2}} \gt \frac{a^{2}+b^{2}}{a+b} <strong>ONLY WHEN</strong> a \gt b \gt 0 Which of the above is/are correct?
- A. 1 <strong>ONLY</strong> ✓
- B. 2 <strong>ONLY</strong>
- C. Both 1 and 2
- D. Neither 1 nor 2
Correct Answer: A. 1 <strong>ONLY</strong>
Explanation
Statement 1 simplifies to \frac{2ab(a-b)}{(a^2+b^2)(a+b)} \gt 0, which requires a-b \gt 0, hence true ONLY when a \gt b \gt 0. Statement 2 simplifies to \frac{ab(a-b)^2}{(a^2+b^2)(a+b)} \gt 0, which is true for ALL a \neq b, a,b \gt 0, not ONLY when a \gt b. Thus, only statement 1 fits the strict condition.
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