Given inequalities:
1.
> where
a>b>0 i.e.,
a and
b are positive and
a>b.
⇒> ⇒> [∵(a−b) is a positive number
] ⇒(a+b)2>a2+b2 ⇒a2+b2+2ab>a2+b2 ⇒2ab>0 which is true.
[∵a>0 and
b>0] 2.
> only when
a>b>0 Since,
a and
b are positive.
⇒(a+b)(a3+b3) ⇒a4+ab3+ba3+b4 ⇒>a4+b4+2a2b2 ⇒ab(b2+a2)−2a2b2>0 ⇒ab[b2+a2−2ab]>0 ⇒ab(a−b)2>0 ∴ If
a and
b are positive then the given inequalities forms into
ab(a−b)2>0, which is true but here, it is not necessary that
a>b. The inequality is true for
b>a>0 also.
i.e.,
a>b>0 is not the only condition.
Hence, option
(a) is correct.
2.
> only when
a>b>0 Since,
a and
b are positive.
∴ If
a and
b are positive then the given inequalities forms into
ab(a−b)2>0, which is true but here, it is not necessary that
a>b.
The inequality is true for
b>a>0 also.
i.e.,
a>b>0 is not the only condition.
Hence, option (a) is correct.