document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=28086>Question 28086</A>: <I><SPAN STYLE=\"word-break: break-all;\"> show that ((x^2y)/2(x+y))+ ((x+y)/2xy)-(x)^1/2 >= 0 for positive real numbers x and y. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #15324 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=15324\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Prove this is true for positive real x & y<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%5E2y%29%2F2%28x%2By%29%29%2B+%28%28x%2By%29%2F2xy%29-%28x%29%5E%281%2F2%29+%3E=+0 ALIGN=MIDDLE  ALT=\"%28%28x%5E2y%29%2F2%28x%2By%29%29%2B+%28%28x%2By%29%2F2xy%29-%28x%29%5E%281%2F2%29+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I think this is the right approach - I admit I'm a little over my head<BR>\n" );
document.write( "If I multiply both sides by 2, the equality stll holds true.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%5E2y%29%2F%28x%2By%29%29%2B+%28%28x%2By%29%2Fxy%29-2%2A%28x%29%5E%281%2F2%29+%3E=+0 ALIGN=MIDDLE  ALT=\"%28%28x%5E2y%29%2F%28x%2By%29%29%2B+%28%28x%2By%29%2Fxy%29-2%2A%28x%29%5E%281%2F2%29+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now if I add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A+%28x%29%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"2%2A+%28x%29%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides, the equality is still true.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%5E2y%29%2F%28x%2By%29%29%2B+%28%28x%2By%29%2Fxy%29+%3E=+2%2A%28x%29%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"%28%28x%5E2y%29%2F%28x%2By%29%29%2B+%28%28x%2By%29%2Fxy%29+%3E=+2%2A%28x%29%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I can rewrite the second term by separating into two fractions<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2By%29%2Fxy ALIGN=MIDDLE  ALT=\"%28x%2By%29%2Fxy\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fy+%2B+1%2Fx ALIGN=MIDDLE  ALT=\"1%2Fy+%2B+1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%5E2y%29%2F%28x%2By%29%29%2B+1%2Fy+%2B+1%2Fx+%3E=+2%2A%28x%29%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"%28%28x%5E2y%29%2F%28x%2By%29%29%2B+1%2Fy+%2B+1%2Fx+%3E=+2%2A%28x%29%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now I choose to multiply both top and bottom of the first term by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28x%2Ay%29 ALIGN=MIDDLE  ALT=\"1%2F%28x%2Ay%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28x+%2F%281%2Fy+%2B+1%2Fx%29%29+%2B+1%2Fy+%2B+1%2Fx+%3E=+2%2A%28x%29%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"+%28x+%2F%281%2Fy+%2B+1%2Fx%29%29+%2B+1%2Fy+%2B+1%2Fx+%3E=+2%2A%28x%29%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now I think I have to let x and y be 0 or very large in all possible combinations<BR>\n" );
document.write( "(a) x = 0 and y = 0<BR>\n" );
document.write( "(b) x = 0 and y approaches infinity<BR>\n" );
document.write( "(c) x approaches infinity and y = 0<BR>\n" );
document.write( "(d) x approaches infinity and y approaches infinity<BR>\n" );
document.write( "That covers the extremes of all real values that x in combination with y can have<BR>\n" );
document.write( " (a) the first term is 0/infinity = 0 and the other terms = infinity<BR>\n" );
document.write( "so this satisfies the equation<BR>\n" );
document.write( " (b) the first term is still 0/infinity<BR>\n" );
document.write( "the other terms are 0 + infinity = infinity, so the equation still holds<BR>\n" );
document.write( " (c) The first term approaches infinity, equation holds<BR>\n" );
document.write( " (d) The first term approaches infinity, equation holds<BR>\n" );
document.write( "Hope this is correct and helps </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );