Question: The arithmeticgeometric inequality By applying the arithmeticgeometric inequality to the first two terms of this inequality, deduce that implies x + y 2 x+y 2

The arithmetic–geometric inequality

implies x + y 2 x+y 2 > xy >xy Use the substitution x = (a + b), y = 1(c + d), where a, b, c and d > 0, to

By applying the arithmetic–geometric inequality to the first two terms of this inequality, deduce that

abcd and hence (a+b+c+d] c + d ) a+b+c+d 4 4 ->Nabcd

implies x + y 2 x+y 2 > xy >xy Use the substitution x = (a + b), y = 2(c + d), where a, b, c and d > 0, to show that and hence that ( + b )( + d ) = (a + b + c + d ) 2 2 4 2 d) = (+b+c+d)* 4 a+b (a + b)(e = (c + 2 2

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