document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1134916>Question 1134916</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find necessary and sufficient conditions on a, b, c and d to ensure that<BR>\n" );
document.write( "f(x) = (ax + b)/(cx + d)\r<BR>\n" );
document.write( "\n" );
document.write( "is a constant function. [Hint: what is the derivative of a constant function?] </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #752431 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=752431\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Algebraically, if the rational function is a constant function, k, then...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ax%2Bb%29%2F%28cx%2Bd%29+=+k ALIGN=MIDDLE  ALT=\"%28ax%2Bb%29%2F%28cx%2Bd%29+=+k\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%2Bb+=+k%28cx%2Bd%29 ALIGN=MIDDLE  ALT=\"ax%2Bb+=+k%28cx%2Bd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%2Bb+=+kcx%2Bkd ALIGN=MIDDLE  ALT=\"ax%2Bb+=+kcx%2Bkd\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+kc ALIGN=MIDDLE  ALT=\"a+=+kc\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+kd ALIGN=MIDDLE  ALT=\"b+=+kd\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This result can be expressed in many ways, including<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fb+=+c%2Fd ALIGN=MIDDLE  ALT=\"a%2Fb+=+c%2Fd\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Using calculus to find the derivative and find the conditions that make the derivative zero....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%28ax%2Bb%29%2F%28cx%2Bd%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28ax%2Bb%29%2F%28cx%2Bd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%2Fdx+=+%28a%28cx%2Bd%29-c%28ax%2Bb%29%29%2F%28%28cx%2Bd%29%5E2%29+=+%28acx%2Bad-acx-bc%29%2F%28%28cx%2Bd%29%5E2%29+=+%28ad-bc%29%2F%28%28cx-d%29%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The condition for the derivative to be zero is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ad-bc=0 ALIGN=MIDDLE  ALT=\"ad-bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "which is equivalent to the earlier forms. </SPAN>\n" );
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