Given that the function f (x) = (MX ^ 2 + 8x + n) / (x ^ 2 + 1) has a range of [1,9], X ∈ R, find the value of M and N?

Given that the function f (x) = (MX ^ 2 + 8x + n) / (x ^ 2 + 1) has a range of [1,9], X ∈ R, find the value of M and N?

It's not easy for me to come up with a method, but if you can, it's a bit too many for you to practice the formula yourself
1. For f (x) = (MX ^ 2 + 8x + n) / (x ^ 2 + 1), two relations between two X 1 x 2 and m n are obtained by Weida's theorem for molecules, which are recorded as Formula 1 and formula 2. At the same time, the discriminant of molecules is written to test m n
2. X 1, x 2 are substituted into f (x) respectively to obtain the other two formulas which are equal to the two maximum values, which are recorded as formula 3 and formula 4
3.1.2.3.4 four equations with four unknowns are established simultaneously. The equations can be solved and may have additional roots. Remember to use the discriminant test