If x is real , then the minimum value of `(x^(2)-3x+4)/(x^(2)+3x+4) ` is :

Options

A. `(0, 1//7)`
B. `(7, oo)`
C. `[1//7, 7]`
D. `[-1//7, 7]`

Answer

C

Explanation

Let `y = (x^(2)-3x + 4)/(x^(2) + 3x + 4)`. Then, `x^(2) (y-1) + 3x(y+1)+4(y-1) = 0`
This equation gives the values of x for given values of y. But, y is the value when x is real. So, the roots of this equaton are real.
`therefore" "9(y+1)^(2) - 16(y-1)^(2) ge 0" "["Using Discriminant" ge 0]`
`rArr" "7y^(2) - 50y + 7 le 0 rArr (7y-1) (y-7) le 0 rArr 1//7 le y le 7`.
Hence, the given expression lies between 1/7 and 7.