If 'x' is real, then greatest value of the expression, `(x+2)/(2x^(2)+3x +6)` is :

Options

A. `1//13`
B. `-1//13`
C. `3`
D. `1//3`

Answer

D

Explanation

If 'x' is real ………..
Let `y = (x+2)/(2x^(2)+3x+6)`
`2x^(2)y + x(3y-1)+(6y-2) =0`
case - I when coefficient of `x^(2) = 0`
`i.e. y = 0`
`x = -2 in R`
case - II when coefficient of `x^(2) != 0`
`:. X` is real
`D ge 0`
`(3y - 1)^(2) - 4xx2yxx(6y-2)`
`-39y^(2)+10y+1ge0`
`-39y^(2)-10y-1le0`
`(3y-1) (13y+1) le 0`
` epsilon [-(1)/(13), (1)/(3)]`
`:.` greatest value `= (1)/(3)`