`f(x)=((x-2)(x-1))/(x-3), forall xgt3`. The minimum value of `f(x)` is equal to

Options

A. `3+2sqrt(2)`
B. `3+2sqrt(3)`
C. `3sqrt(2)+2`
D. `3sqrt(2)-2`

Answer

A

Explanation

Let
`x - 3 = t`
`implies x - 2 (t + 1)` and `x - 1 = t + 2`
`implies f(x) = ((x - 2) (x - 1))/(t)`
`= ((t + 1) (t + 2))/(t)`
`= ((t^(2) + 3t 2))/(t)`
`t + (2)/(t) + 3`
`le 3 + 2 sqrt(2)` (using A.M `ge` G.M as `t gt 0`)