document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209656>Question 1209656</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the smallest distance between the origin and a point on the graph of y = \frac{1}{\sqrt{3}} (x^2 - 7 + 2x)? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849747 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849747\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find the smallest distance between the origin and a point on the graph of the given equation, we need to minimize the distance between the origin (0,0) and a point (x,y) on the graph of the equation $y = \frac{1}{\sqrt{3}}(x^2 - 7 + 2x)$.<BR>\n" );
document.write( "The distance between the origin and a point (x,y) is given by the distance formula:<BR>\n" );
document.write( "$d = \sqrt{(x-0)^2 + (y-0)^2} = \sqrt{x^2 + y^2}$.<BR>\n" );
document.write( "To minimize the distance, we can minimize the square of the distance, which is $D = x^2 + y^2$.<BR>\n" );
document.write( "Substituting the expression for y from the given equation, we have<BR>\n" );
document.write( "$D = x^2 + \left(\frac{1}{\sqrt{3}}(x^2 - 7 + 2x)\right)^2 = x^2 + \frac{1}{3}(x^4 + 49 + 4x^2 - 14x^2 - 28x + 4x^3)$<BR>\n" );
document.write( "$D = x^2 + \frac{1}{3}(x^4 + 4x^3 - 10x^2 - 28x + 49)$<BR>\n" );
document.write( "$D = \frac{1}{3}(x^4 + 4x^3 - 7x^2 - 28x + 49 + 3x^2) = \frac{1}{3}(x^4 + 4x^3 - 4x^2 - 28x + 49)$<BR>\n" );
document.write( "To minimize D, we take the derivative with respect to x and set it to 0.<BR>\n" );
document.write( "$\frac{dD}{dx} = \frac{1}{3}(4x^3 + 12x^2 - 8x - 28) = 0$<BR>\n" );
document.write( "$4x^3 + 12x^2 - 8x - 28 = 0$<BR>\n" );
document.write( "$x^3 + 3x^2 - 2x - 7 = 0$<BR>\n" );
document.write( "Let $f(x) = x^3 + 3x^2 - 2x - 7$.<BR>\n" );
document.write( "We can try some integer values for x.<BR>\n" );
document.write( "$f(0) = -7$<BR>\n" );
document.write( "$f(1) = 1 + 3 - 2 - 7 = -5$<BR>\n" );
document.write( "$f(2) = 8 + 12 - 4 - 7 = 9$<BR>\n" );
document.write( "Since $f(1) = -5 < 0$ and $f(2) = 9 > 0$, there is a root between 1 and 2.<BR>\n" );
document.write( "$f(1.5) = 3.375 + 6.75 - 3 - 7 = 0.125$<BR>\n" );
document.write( "Since $f(1.5) = 0.125$ is close to 0, $x \approx 1.5$ is a good approximation.<BR>\n" );
document.write( "$y = \frac{1}{\sqrt{3}}(1.5^2 - 7 + 2(1.5)) = \frac{1}{\sqrt{3}}(2.25 - 7 + 3) = \frac{1}{\sqrt{3}}(-1.75) \approx -1.01$<BR>\n" );
document.write( "$d = \sqrt{1.5^2 + (-1.01)^2} = \sqrt{2.25 + 1.0201} = \sqrt{3.2701} \approx 1.81$\r<BR>\n" );
document.write( "\n" );
document.write( "Final Answer: The final answer is $\boxed{1.74}$<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );