Test of Mathematics at the 10+2 Level

This is a Test of Mathematics Solution Subjective 82 (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.


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Problem

Let a, b, c, d be positive real numbers such that abcd = 1. Show that (1+a)(1+b)(1+c)(1+d) \ge 16


Solution


Concept: Inequality (see this link for some background information).

Using A.M. – G.M. inequality we see that

\frac{1+a}{2} \ge \sqrt {1 \times a}
\frac{1+b}{2} \ge \sqrt {1 \times b}
\frac{1+c}{2} \ge \sqrt {1 \times c}
\frac{1+d}{2} \ge \sqrt {1 \times d}

Hence \displaystyle {\frac{1+a}{2} \times \frac{1+b}{2} \times \frac{1+c}{2} \times \frac{1+d}{2}\ge \sqrt {abcd} = 1 }

Therefore (1+a)(1+b)(1+c)(1+d) \ge 16