Question 1134916: Find necessary and sufficient conditions on a, b, c and d to ensure that
f(x) = (ax + b)/(cx + d)
is a constant function. [Hint: what is the derivative of a constant function?]
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source):
Answer by ikleyn(52784)  (Show Source):
You can put this solution on YOUR website! .
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,  =  .
It is precisely the same condition as proportionality of the pairs (a,b) and (c,d).
Solved and answered.
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