Inequality Problem

Given : $ { x }^{ 2 }+{ y }^{ 2 }=1 $

Now prove that: $\frac { 1 }{ 1+{ x }^{ 2 } } +\frac { 1 }{ 1+{ y }^{ 2 } } +\frac { 1 }{ 1+xy } \ge \frac { 3 }{ 1+{ \left( \frac { x+y }{ 2 } \right) }^{ 2 } }$

#Algebra #Inequalities

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

put x and y as asin@ and acos@ and then solve it

Abhishek Sharma - 5 years, 7 months ago

Solve without using any kind of Trigonometrical topics and Jensen's Inequality. Use: AM, GM, HM, CS, Titu's Lemma or any other kind of inequality to solve.

Jolly Ghosh - 5 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$