document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=984855>Question 984855</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the points on inflection and discuss the concavity of the function:<BR>\n" );
document.write( " f(x)=x sqrt (x+8) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #605878 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=605878\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f(x) = x * sqrt(x+8)<BR>\n" );
document.write( "we will take the first derivative of f(x) and set it equal to 0 to find the points of inflection<BR>\n" );
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document.write( "to find the first derivative, we will use the product rule, namely<BR>\n" );
document.write( "d(uv)/dx = u'v + v'u<BR>\n" );
document.write( "let u = x and v = (x+8)^(1/2), then<BR>\n" );
document.write( "u' = 1<BR>\n" );
document.write( "v' = (1/2) * (x+8)^(-1/2)<BR>\n" );
document.write( "d(uv)/dx = (x+8)^(1/2) + (x/2) * (x+2)^(-1/2)<BR>\n" );
document.write( "d(uv)/dx = f'(x) = (3x + 16) / (2*(x+8)^(1/2))<BR>\n" );
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document.write( "the point of inflection is<BR>\n" );
document.write( "(3x + 16) / (2*(x+8)^(1/2)) = 0<BR>\n" );
document.write( "3x + 16 = 0<BR>\n" );
document.write( "x = -16 / 3<BR>\n" );
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document.write( "the x axis intercepts for f(x) are<BR>\n" );
document.write( "f(x) = 0 = x * sqrt(x+8)<BR>\n" );
document.write( "x = 0 and x = -8<BR>\n" );
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document.write( "In order to discuss concavity of f(x), we calculate the second derivative of f(x), we do this by starting with f'(x) and taking its derivative<BR>\n" );
document.write( "f'(x) = (3x + 16) / (2*(x+8)^(1/2))<BR>\n" );
document.write( "to take this derivative, we use the quotient rule, namely<BR>\n" );
document.write( "d(u/v)/dx = (u'v - uv') / v^2<BR>\n" );
document.write( "u = (3x + 16)<BR>\n" );
document.write( "v = (2 *(x+8)^(-1/2))<BR>\n" );
document.write( "u' = 3<BR>\n" );
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document.write( "for v' we use the product rule<BR>\n" );
document.write( "u = 2<BR>\n" );
document.write( "v = (x+8)^(-1/2)<BR>\n" );
document.write( "v' = 0 + (2 * (x+8)^(-3/2) / -2<BR>\n" );
document.write( "v' =  -(x+8)^(-3/2)<BR>\n" );
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document.write( "d(u/v)/dx = f''(x) = (3*(2 *(x+8)^(-1/2))) + ((3x + 16)*(x+8)^(-3/2)) / (4*(x+8))<BR>\n" );
document.write( "f''(x) = (3x + 32) / (4*(x+8)^(3/2))<BR>\n" );
document.write( "to discuss concavity, we evaluate f''(x) on the interval (-8, +infinity)<BR>\n" );
document.write( "note that for x < -8, f(x) is imaginary<BR>\n" );
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document.write( "from inspection we see that f''(x) is positive on (-8, infinity) which means that f(x) is concave up on that interval, here is the graph of f(x)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-8%2C+10%2C+-15%2C+10%2C+x+%2A+sqrt%28x%2B8%29%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-8%2C+10%2C+-15%2C+10%2C+x+%2A+sqrt%28x%2B8%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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