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Given the equation $e^x=\frac{bx-a}{c+dx}$ where : $a,b,c,d \in R$ and $a+c=-1$, how can I prove that there are at most two real solutions? Graphically it's pretty clear to see since there is an exponential and a hyperbole, but formally how one should prove this? Many thanks.

EDIT : I am looking for positive solutions only.

EDIT : Prove a convex and concave function can have at most 2 solutions here I’ve found the answer to my question.

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  • $\begingroup$ Here's a related question: how many intersections can a convex and a concave function have? Why? How about two convex functions or two concave functions? $\endgroup$ Commented Dec 19, 2022 at 16:51
  • $\begingroup$ math.stackexchange.com/questions/2112108/… Check here to find more. $\endgroup$ Commented Dec 19, 2022 at 20:10
  • $\begingroup$ It would be a fun proof to put both graphs on a sphere. I think stereographic projection will turn hyperbolas into ellipses. $\endgroup$ Commented Dec 20, 2022 at 3:38
  • $\begingroup$ Just FYI - "hyperbole" is wildly exaggerated speech, while a "hyperbola" is a conic section. "The most incredibly beautiful double-curve every produced by slicing a cone with a plane" would be a example of both. $\endgroup$ Commented Dec 20, 2022 at 13:21

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