document.write( "Question 1120745: We may define the three means (arithmetic, geometric, and harmonic) of two positive numbers a and b as\r
\n" ); document.write( "\n" ); document.write( "A = (a+b)/2\r
\n" ); document.write( "\n" ); document.write( "G = √ab\r
\n" ); document.write( "\n" ); document.write( "H = 2ab/ a+b, respectively\r
\n" ); document.write( "\n" ); document.write( "a. Show the inequality, A ≥ G ≥ H, holds if and only if a = b \n" ); document.write( "
Algebra.Com's Answer #736716 by ikleyn(52803)\"\" \"About 
You can put this solution on YOUR website!
.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "                1.   Prove  A >= G  holds if and only if   a = b.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "You have this chain of equivalent inequalities\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\"%28a%2Bb%29%2F2\" >= \"sqrt%28ab%29\"  <------>  a + b >= \"2%2Asqrt%28ab%29\"  <------>  \"%28sqrt%28a%29%29%5E2\" + \"%28sqrt%28b%29%29%5E2\"  >= \"2%2Asqrt%28a%29%2Asqrt%28b%29\"  <------>  \"%28sqrt%28a%29%29%5E2+-+2%2Asqrt%28a%29%2Asqrt%28b%29+%2B+%28sqrt%28b%29%5E2%29\" >= 0  <------>  \"%28sqrt%28a%29+-+sqrt%28b%29%29%5E2\" >= 0\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It is true for all real non-negative  numbers  \"a\" and \"b\",   and the equality is hold if and only if   \"sqrt%28a%29\" - \"sqrt%28b%29\" = 0, \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "which in turn is equivalent to  a = b.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "You can read this chain of inequalities / (of arguments) from the right to the left - then you will get the proof you need.\r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "                2.   Prove  G >= H  holds if and only if   a = b.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "\"sqrt%28ab%29\" >= \"%282ab%29%2F%28a%2Bb%29\"  <------>  \"a\" + \"b%29\" >= \"%282ab%29%2Fsqrt%28ab%29\"  <------>  \"a\" + \"b%29\" >= \"2%2Asqrt%28ab%29\",\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "and the last inequality is exactly  THE SAME  as the proven in  #1.\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );