Test of Mathematics Solution Subjective 80 - Inequality of squares

This is a Test of Mathematics Solution Subjective 80 (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

If ${a, b, c}$ are positive numbers, then show that
$\displaystyle{{\frac{b^2 + c^2}{b + c}} + {\frac{c^2 + a^2}{c + a}} + {\frac{a^2 + b^2}{a + b}} \ge {a + b + c}}$ .

Solution

We know that ${a, b, c > 0}$

L.H.S = ${\displaystyle{{\frac{b^2 + c^2}{b + c}} + {\frac{c^2 + a^2}{c + a}} + {\frac{a^2 + b^2}{a + b}}}}$

${\ge}$ ${\displaystyle{\frac{{\frac{b^2 + c^2 + 2bc}{b + c}} + {\frac{c^2 + a^2 + 2ac}{c + a}} + {\frac{a^2 + b^2 + 2ab}{a + b}}}{2}}}$ [ as ${b + c}$ , ${c + a}$ , ${a + b > 0}$ & ${x^2 + y^2 > 2xy}$ for all ${x + y {\in} |R}$ ]

= ${a + b + c}$
= R.H.S [ proved ]

Related

This is a Test of Mathematics Solution Subjective 80 (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

If ${a, b, c}$ are positive numbers, then show that
$\displaystyle{{\frac{b^2 + c^2}{b + c}} + {\frac{c^2 + a^2}{c + a}} + {\frac{a^2 + b^2}{a + b}} \ge {a + b + c}}$ .

Solution

We know that ${a, b, c > 0}$

L.H.S = ${\displaystyle{{\frac{b^2 + c^2}{b + c}} + {\frac{c^2 + a^2}{c + a}} + {\frac{a^2 + b^2}{a + b}}}}$

${\ge}$ ${\displaystyle{\frac{{\frac{b^2 + c^2 + 2bc}{b + c}} + {\frac{c^2 + a^2 + 2ac}{c + a}} + {\frac{a^2 + b^2 + 2ab}{a + b}}}{2}}}$ [ as ${b + c}$ , ${c + a}$ , ${a + b > 0}$ & ${x^2 + y^2 > 2xy}$ for all ${x + y {\in} |R}$ ]

= ${a + b + c}$
= R.H.S [ proved ]

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