Question 1134916
.
<pre>
From the first glance to the problem, the answer is obvious:


    The necessary and sufficient condition is proportionality of the pairs (a,b) and (c,d).


To get a formal proof, you can take the derivative of the function.


The numerator of the derivative function then is 


    a*(cx + d) - (ax + b)*c = (ac - ac)*x + ad - bc = ad - bc.


The function (the original rational function / (fraction) ) is a constant if and only if the numerator of the derivative is identically zero, 

which leads to the equality  ad - bc = 0.


In turn, it means that  ad = bc,  or, equivalently,  {{{a/b}}} = {{{c/d}}}.


It is precisely the same condition as proportionality of the pairs (a,b)  and  (c,d).
</pre>

Solved and answered.