This is a Test of Mathematics Solution Subjective 83 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

*If a and b are positive real numbers such that a + b = 1, prove that + \( \left( b + \frac{1}{b} \right)^2 \ge \frac{25}{2} \) *

We replace 1 by a+b and expand the squares.

Now we use A.M. – G.M. inequality to have

Also we apply the square mean inequality to have

Since a+b = 1, we have

Combining all of them we have

**Discussion**