I have solved this system of equations (see below) in Mathematica for real x where the coefficients of the equations are functions of real parameters a,b and c. Mathematica then displays real solutions x with constraints on a,b and c.
The constraints for c (for example) are written in function of roots objects Root[,k]. In the output, I see for instance Root[,1] < c <= Root[,2]. On the other hand, I also see the condition 0< c < Root[,3].
If I'm correct, this implies that I can assume that Root[,1] < Root[,2]? However, can I also assume that Root[,2] < Root[,3]? Furthermore, since Mathematica displays the constraints this way I can assume that these roots (I mean the root objects) are all real, otherwise the statements would be meaningless? I know these root objects are difficult to handle but I really need a proper interpretation to set up the admissible (a,b,c) domain such that the system admits a real solution x.
The Mathematica code for the system is:
Reduce[
16 x^4 - 40 a x^3 + (15 a^2 + 24 b) x^2 - 18 a b x + 3 b^2 == 0
&& 5 a x - 4 x^2 - b > 0
&& 15 a x - 20 x^2 - 3 b < 0
&& 4 x^3 - 8 c x^2 + 5 c a x - c b > 0
&& c > 0 && x > 0,
x, Reals]
Thanks in advance! Cheers.
