Based on this answer, I tried to find the condition to function $y = \dfrac{a x+ b}{c x +d}$ where $ad \neq 0$ and $a d - b c \neq 0$ increasing on $(-d/c, +\infty)$. I tried
Clear["Global`*"];
f[x_] = (a x + b)/(c x + d );
Reduce[ForAll[x, x > -d/c, Derivative[1][f][x] > 0], {a, b, c, d}]
I know that, the answer is $a d - bc > 0$. How can I get this result?
Reduce
is oriented toward solving for variables, not expressions. So it determines the solutions for three cases fora
, namelya<0
,a==0
, anda>0
. It's hard to put those cases back together, since those cases are solved forb
,c
,d
in turn in a similar way resulting in their own sub-cases. I doubt this is what you want but it gives the desired result:Or @@ Select[! Eliminate[Derivative[1][f][x] == 0, {x}] /. {{Unequal -> Less}, {Unequal ->Greater}}, Reduce[Implies[#, ForAll[x, x != -d/c, Derivative[1][f][x] > 0]], Reals] &]
$\endgroup$