document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1032808>Question 1032808</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider<BR>\n" );
document.write( "f(x) = x sqrt(x + 2) and g(x) = x/ sqrt(5 &#8722; x)<BR>\n" );
document.write( ".(a) Find the domains of the functions f and g.<BR>\n" );
document.write( "(b) Find the derivatives of the functions f and g in part (a).<BR>\n" );
document.write( "(c) Find the values of x so that (i) f'(x) = 0, (ii) g'(x) = 0<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #647413 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=647413\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Domain of x(sqrt(x+2)) are all x greater than or equal to -2. Any x less than that gives sqrt (number <0)<BR>\n" );
document.write( "domain of x/sqrt(5-x) is x less than 5.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C200%2C-10%2C10%2C-10%2C10%2Cx%2Asqrt%28x%2B2%29%2Cx%2F%28sqrt%285-x%29%29%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C200%2C-10%2C10%2C-10%2C10%2Cx%2Asqrt%28x%2B2%29%2Cx%2F%28sqrt%285-x%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "f(x)=x*(x+2)^(1/2)<BR>\n" );
document.write( "f'(x)=x*(1/2)(x+2)^(-1/2)+(x+2)^(1/2)<BR>\n" );
document.write( "set that equal to 0.<BR>\n" );
document.write( "sqrt(x+2)=-x(1/2sqrt(x+2))<BR>\n" );
document.write( "square both sides<BR>\n" );
document.write( "x+2=x^2/4(x+2)<BR>\n" );
document.write( "4(x+2)(x+2)=x^2<BR>\n" );
document.write( "4x^2+16x+16=x^2<BR>\n" );
document.write( "3x^2+16x+16=0<BR>\n" );
document.write( "(3x+4)(x+4)=0<BR>\n" );
document.write( "x=-4/3.-4, but -4 doesn't work in original derivative. <BR>\n" );
document.write( "x=-4/3 for derivative = 0. Notice that the tangent to the function at -4/3 has 0 slope.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C200%2C-2%2C1%2C-2%2C1%2Cx%2Asqrt%28x%2B2%29%2Cx%2F%28sqrt%285-x%29%29%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C200%2C-2%2C1%2C-2%2C1%2Cx%2Asqrt%28x%2B2%29%2Cx%2F%28sqrt%285-x%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "==============================<BR>\n" );
document.write( "g(x)=x*(5-x)^(1/2)<BR>\n" );
document.write( "g'(x)=x*(1/2)(5-x)^(-1/2)+(5-x)^(1/2)<BR>\n" );
document.write( "set it equal to 0<BR>\n" );
document.write( "0=x(1/2)(5-x)^(-1/2)+(5-x)^(1/2)<BR>\n" );
document.write( "-(5-x)^(1/2)=x*(1/2)(1/sqrt(5-x))<BR>\n" );
document.write( "square both sides<BR>\n" );
document.write( "5-x=x^2(1/4(5-x))<BR>\n" );
document.write( "4(5-x)^2=x^2<BR>\n" );
document.write( "4x^2-40x+100=x^2<BR>\n" );
document.write( "3x^2-40x+100=0<BR>\n" );
document.write( "(3x-10)(x-10)=0<BR>\n" );
document.write( "=10/3,10<BR>\n" );
document.write( "But x cannot equal these values, so first derivative will not ever equal 0. In the graph, there is no place where the slope of the tangent line is 0.\r<BR>\n" );
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