Question 1134916
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Algebraically, if the rational function is a constant function, k, then...<br>
{{{(ax+b)/(cx+d) = k}}}
{{{ax+b = k(cx+d)}}}
{{{ax+b = kcx+kd}}}
{{{a = kc}}} and {{{b = kd}}}<br>
This result can be expressed in many ways, including<br>
{{{a/b = c/d}}}<br>
Using calculus to find the derivative and find the conditions that make the derivative zero....<br>
{{{f(x) = (ax+b)/(cx+d)}}}
{{{df/dx = (a(cx+d)-c(ax+b))/((cx+d)^2) = (acx+ad-acx-bc)/((cx+d)^2) = (ad-bc)/((cx-d)^2)}}}<br>
The condition for the derivative to be zero is<br>
{{{ad-bc=0}}}<br>
which is equivalent to the earlier forms.