document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209306>Question 1209306</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let x and y be nonnegative real numbers.  If xy = 4/3, then find the minimum value of 2x + 6y + x^2 + 3y^2.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848182 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848182\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy+=+4%2F3 ALIGN=MIDDLE  ALT=\"xy+=+4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > for y to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+4%2F%283x%29 ALIGN=MIDDLE  ALT=\"y+=+4%2F%283x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Then we plug it into the expression to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+%2B+6y+%2B+x%5E2+%2B+3y%5E2 ALIGN=MIDDLE  ALT=\"2x+%2B+6y+%2B+x%5E2+%2B+3y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+%2B+6%2A%284%2F%283x%29%29+%2B+x%5E2+%2B+3%2A%284%2F%283x%29%29%5E2 ALIGN=MIDDLE  ALT=\"2x+%2B+6%2A%284%2F%283x%29%29+%2B+x%5E2+%2B+3%2A%284%2F%283x%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%5E4%2B6x%5E3%2B24x%2B16%29%2F%283x%5E2%29 ALIGN=MIDDLE  ALT=\"%283x%5E4%2B6x%5E3%2B24x%2B16%29%2F%283x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The goal is to make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%283x%5E4%2B6x%5E3%2B24x%2B16%29%2F%283x%5E2%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%283x%5E4%2B6x%5E3%2B24x%2B16%29%2F%283x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as small as possible.<BR>\n" );
document.write( "Note that x = 0 isn't allowed in the domain so we're focused on the interval x > 0.\r<BR>\n" );
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document.write( "You can use differential calculus or you can use a graphing calculator to determine the local minimum when x > 0\r<BR>\n" );
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document.write( "Two recommended graphing tools are <a  rel=nofollow HREF=\"https://www.desmos.com/calculator\">Desmos</a> and <a  rel=nofollow HREF=\"https://www.geogebra.org/calculator\">GeoGebra</a><BR>\n" );
document.write( "There are many others to pick from.\r<BR>\n" );
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document.write( "Here's the Desmos graph<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/nieyc6olhx\">https://www.desmos.com/calculator/nieyc6olhx</a><BR>\n" );
document.write( "When focusing on the interval x > 0 the lowest part of the red curve is approximately at (1.65132, 12.82996)<BR>\n" );
document.write( "This is marked as the green point on the curve. \r<BR>\n" );
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document.write( "The y coordinate of this local minimum is the smallest value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and also the smallest value of the original expression.\r<BR>\n" );
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document.write( "Side note: be sure not to mix up the y coordinate of the local minimum with the \"y\"s in either xy = 4/3 or 2x + 6y + x^2 + 3y^2. Those are different y values. \r<BR>\n" );
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document.write( "Answer: <font color=red>12.82996 (approximate)</font><BR>\n" );
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