document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163545>Question 1163545</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the intervals of concavity and state any points of inflection.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29+=+%281%2F6%29x%5E4+-+%287%2F6%29x%5E3+%2B+%283%2F2%29x%5E2+-+8x+%2B+10+ ALIGN=MIDDLE  ALT=\"+f%28x%29+=+%281%2F6%29x%5E4+-+%287%2F6%29x%5E3+%2B+%283%2F2%29x%5E2+-+8x+%2B+10+\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #787668 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=787668\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x) = \frac{1}{6}x^4 - \frac{7}{6}x^3+\frac{3}{2}x^2-8x+10\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f'(x) = \frac{1}{6}*4x^3 - \frac{7}{6}*3x^2+\frac{3}{2}*2x-8\"> First derivative\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f'(x) = \frac{4}{6}x^3 - \frac{21}{6}x^2+\frac{6}{2}x-8\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f'(x) = \frac{2}{3}x^3 - \frac{7}{2}x^2+3x-8\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f^{\prime \prime}(x) = \frac{2}{3}*3x^2 - \frac{7}{2}*2x+3\"> Second derivative\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f^{\prime \prime}(x) = 2x^2 - 7x+3\"> \r<BR>\n" );
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document.write( "Solve <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f^{\prime \prime}(x) = 0\"> for x to find the two solutions x = 1/2 and x = 3\r<BR>\n" );
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document.write( "Plug in a value to the left of x = 1/2, say x = 0, to find that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f^{\prime \prime}(0)\"> is a positive value (the result is positive 3 when you use x = 0), so <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f^{\prime \prime}(x)\"> is positive on the interval <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (-\infty, 1/2)\"> meaning f(x) is concave up on this interval.\r<BR>\n" );
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document.write( "Plug in some number between x = 1/2 and x = 3. I'll use x = 1. If you plug x = 1 into the second derivative function, you should get a negative result. This means <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f^{\prime \prime}(x)\"> is negative throughout the interval <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (1/2, 3)\"> and f(x) is concave down on this interval.\r<BR>\n" );
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document.write( "Lastly, f(x) is concave up on the interval <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (3, \infty)\"> because plugging in a value to the right of x = 3, say x = 4, leads to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f^{\prime \prime}(x)\"> being positive.\r<BR>\n" );
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document.write( "Because there is a sign change of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f^{\prime \prime}(x)\"> as we pass through the roots x = 1/2 and x = 3, this means we have points of inflection at these x values. Plug those x values one at a time into f(x) to find the corresponding y coordinates are y = 6.23958 (approx) and y = -18.5\r<BR>\n" );
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document.write( "The points of infection are (1/2, 6.23958) and (3, -18.5) </SPAN>\n" );
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