Consider the following inequalities:

1. \(\frac{a^2 - b^2 }{a^2 + b^2 } > \frac{a - b}{a + b} \) where a > b > 0

2. \(\frac{a^3 + b^3 }{a^2 + b^2 } > \frac{a^2 + b^2}{a + b} \) only when a > b > 0

Which of the above is/are correct?

Calculation:

Statement:1    \(\frac{a^2 - b^2 }{a^2 + b^2 } > \frac{a - b}{a + b} \) where a > b > 0

Let a = 2 and b = 1

⇒ \(\frac{2^2 - 1^2 }{2^2 + 1^2 } > \frac{2 - 1}{2 + 1} \)

⇒ \(\frac{3 }{5 } > \frac{1}{3} \) which is correct.

Hence, statement 1 is correct.

Statemnt:2   \(\frac{a^3 + b^3 }{a^2 + b^2 } > \frac{a^2 + b^2}{a + b} \) only when a > b > 0

Let a = 2 and b = 1

⇒ \(\frac{2^3 + 1^3 }{2^2 + 1^2 } > \frac{2^2 + 1^2}{2 + 1} \)

⇒ \(\frac{9 }{5 } > \frac{5}{3} \) Which is also correct.

This inequality also holds for a = 1 and b = 2 i.e.

Statement 2 is correct for b > a > 0 as well.

But according to statement 2, inequality will be correct only if a > b > 0

So, statement 2 is incorrect.

Hence, statement (1) is correct but (2) statement (2) is incorrect.

Mistake PointsPlease note the language of  the second statement

"only when a > b > 0"

We have proved thet, statement is correct for 

a = 2 & b = 1 (i.e. a > b > 0) and b = 2 & a = 1 (i.e. b > a > 0)

Hence, due to only word, statement 2 will become incorrect.