SOLUTION: 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?]

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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): You can put this solution on YOUR website!


Algebraically, if the rational function is a constant function, k, then...




and

This result can be expressed in many ways, including



Using calculus to find the derivative and find the conditions that make the derivative zero....




The condition for the derivative to be zero is



which is equivalent to the earlier forms.
Answer by ikleyn(52786)   (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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