Test of Mathematics Solution Subjective 82 - Inequality on four positive real numbers

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.

Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta

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

Related

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.

Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta

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

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