1.

Show that the expression `(px^2 + 3x - 4)/( p + 3x - 4x^2)` will be capable of all values when x is real,provided that p has any value between 1 and 7.

Answer» `(px^2+3x-4)/(p+3x-4x^2)=y`
`px^2+3x-4=py+3xy-4yx^2`
`(4y+p)x^2+(3-3y)x-4-py=0`
`D>=0`
`9(1-y)^2+4(4+py)(4y+p)>=0`
`8y^2-18y+9+64y+16p+16py^2+4p^2y>=0`
`(9+16P)y^2+(46+4p^2)y+9+16p>=0`
`D<=0`
`(46+4p^2)^2-4(9+16p)^2<=0`
`4[(23+2p^2)^2-(9+16p)^2]<=0`
`(p^2+8p+16)(p^2-8p+7)<=0`
`D<0`
`p^2-8p+7<=0`
`(p-1)(p-7)<=0`
`P in [1,7]`.


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