`f: RvecR ,f(x)(3x^2+m x+n)/(x^2+1)dot` If the range of this function is `[-4,3)` , then filnd the value of `|m+n|` is_________.

Answer

4

Explanation

`y=(3x^(2)+mx+n)/(x^(2)+1)`
or `x^(2)(y-3)-mx+y-n=0`
Since `x in R`
`Dge0`
`rArrm^(2)-4(y-3)(y-n)ge0`
or `m^(2)=4(y^(2)-ny-3y+3n)ge0`
`4y^(2)-4y(n+3)+12n-m^(2)le0" "(1)`
Also given `(y + 4)(y - 3) le 0 `
`y^(2)+y-12 le 0" "(2)`
Comparing (1) and (2) , we get `(4)/(1)=-(4(n+3))/(1)=(12n-m^(2))/(-12)`
`rArr m = 0 and n =-4`