document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=382720>Question 382720</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What point on the graph of the equation y=x^2 is closest to the point (3,1)? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #271154 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=271154\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We want to minimize the distance between the points (x, x^2) and (3,1). Using the distance formula, this distance z is equal to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+sqrt+%28%28x-3%29%5E2+%2B+%28x%5E2+-+1%29%5E2%29+=+sqrt%28x%5E4+-+x%5E2+-+6x+%2B+10%29 ALIGN=MIDDLE  ALT=\"z+=+sqrt+%28%28x-3%29%5E2+%2B+%28x%5E2+-+1%29%5E2%29+=+sqrt%28x%5E4+-+x%5E2+-+6x+%2B+10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We wish to minimize the value of z. By taking the derivative of z and setting it to zero, we obtain\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dz%2Fdx+=+%284x%5E3+-+2x+-+6%29%2F%282%2Asqrt%28x%5E4+-+x%5E2+-+6x+%2B+10%29%29+=+%282x%5E3+-+x+-+3%29%2F%28sqrt%28x%5E4+-+x%5E2+-+6x+%2B+10%29%29+=+0 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This occurs when\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3+-+x+-+3+=+0 ALIGN=MIDDLE  ALT=\"2x%5E3+-+x+-+3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the denominator is nonzero.\r<BR>\n" );
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document.write( "I had to use a calculator to determine that the only real root is x = 1.289623 (meaning that a critical point is located there, it can be checked that it minimizes the distance). Therefore the closest point on the graph y = x^2 to (3,1) is (1.289623, 1.663127) </SPAN>\n" );
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