document.write( "Question 1143014: If x+y+z=16 then find the maximum value of (x-3)(y-5)(z-2). Given that (x-3)>0, (y-5)>0, (z-2)>0 \n" ); document.write( "

Algebra.Com's Answer #763789 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "

\r\n" );
document.write( "By analogy with the well known AM-GM inequality (\"Arithmetic Mean - Geometric Mean inequality\") for two variables \"a\" and \"b\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    ab <= ,         (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "there is AM-GM inequality for three variables \"a\", \"b\" and \"c\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    abc <= .      (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Inequalities (1) and (2) are valid for any two and three variables, respectively, that are real non-negative numbers.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Apply inequality (2), taking  \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    a = x-3,  b = y-5,  c = z-2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "You will get\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    (x-3)*(y-5)*(z-2) <=  =  =  =  =  = 8.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus for any 3 values of x, y and z, restricted by the equality\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    x + y + z = 16   and  inequalities  x >= 3, y >= 5  and  z >= 2,\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "the inequality  \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    (x-3)*(y-5)*(z-2) <= 8 \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "is held.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From the other side, at x= 5, y= 7  and z= 4 we have\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    (x-3)*(y-5)*(z-2) = (5-3)*(7-5)*(4-2) = 2*2*2 = 8.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "and the values of x, y and z satisfy all needed restrictions.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus the maximum value of (x-3)*(y-5)*(z-2), where  x, y and z are restricted by \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    x + y + z = 16,    x >= 3, y >= 5  and  z >= 2\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "is 8.        ANSWER\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "--------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "S O L V E D.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );