document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194170>Question 1194170</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The positive variables x and y are such that x^4 y = 32. A third variable z id defined by z = x^2 + y.<BR>\n" );
document.write( "Find the values x and y that give z a stationary value and show that this value of z is a minimum. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #826311 by </B><A HREF=/tutors/aboutme.mpl?userid=htmentor><B>htmentor(1343)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=htmentor><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=htmentor><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=826311\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> x^4 y = 32 -> y = 32/x^4<BR>\n" );
document.write( "Thus z = x^2 + y -> z = x^2 + 32/x^4<BR>\n" );
document.write( "Stationary values are the points where dz/dx = 0:<BR>\n" );
document.write( "dz/dx = 0 = 2x - 128/x^5<BR>\n" );
document.write( "To solve for x, multiply through by x^5:<BR>\n" );
document.write( "2x^6 - 128 = 0 -> x^6 = 64<BR>\n" );
document.write( "Solutions: x = 2, -2<BR>\n" );
document.write( "Thus y = 32/(2)^4 = 2<BR>\n" );
document.write( "The stationary points are (2,2) and (-2,2)<BR>\n" );
document.write( "Computing the 2nd derivative, since d^2z/dx^2 = 2 + 640/x^6 is positive at x=2, -2, the points are minima<BR>\n" );
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