function f , R `->`R , `f(x) = (3x^2+mx+n)/(x^2+1)` , if the range of function is [-4,3), find the value of |m+n| is .........

Answer

4

Explanation

`f:RtoR`
`f(x)=(3x^(2)+mx+n)/(x^(2)+1)`
`y=(3x^(2)+mx+n)/(x^(2)+1)`
`(y-3)x^(2)-mx+(y-n)=0`
x must be real
`=m^(2)-4(y-3)(y-n)ge0`
`=4(y-3)(y-n)-m^(2)le0`
`=(y-3)(y-n)-(m^(2))/(4)le0`
As range in `[-4,3]`
`y^(2)-(3+n)y+3n-(m^(2))/(4)=(y+4)(y-3)=y^(2)+y-12`
`=(-3+n)=1impliesn=-4`
`=3n-(m^(2))/(4)=-12`
`=m=0`
`=y^(2)-(3+n)y+3n-(m^(2))/(4)=(y+4)(y-3)-y^(2)+y-12`
`=-(3+n)=1impliesn-4`
`3n-(m^(2))/(4)=-12`
`m=0`