Let `f(x)=ax^(2)+bx+c`, `a ne 0`, `a`, `b`, `c in I`. Suppose that `f(1)=0`, `50 lt f(7) lt 60 ` and `70 lt f(8) lt 80`.
Number of integral values of `x` for which `f(x) lt 0` is

Options

A. `0`
B. `1`
C. `2`
D. `3`

Answer

B

Explanation

`f(x)=ax^(2)+bx+c`, `a ne 0`, `a,b,c in I`
`f(1)=0` ……..`(i)`
`implies a+b+c=0`
`50 lt f(7) lt 60`
`50 lt 49a+7b+c lt 60`
`implies 50 lt 48a+6b lt 60`
`implies (50)/(6) lt 8a+b lt 10`
`implies 8a+b=9`……`(ii)`
Also `70 lt f(8) lt 80`
`implies 70 lt 64a+8b+c lt 80`
`implies70 lt 63a+7b lt 80`
`implies 10 lt 9a+b lt (80)/(7)`
`implies 9a+b=11`.........`(iii)`
From `(i)`, `(ii)` and `(iii)`, `a=2`, `b=-7`, `c=5`
`implies f(x)=2x^(2)-7x+5=(2x-5)(x-1)`
`=2(x^(2)-(7)/(2)x+(5)/(2))`
`=2((x-(7)/(4))^(2)-(9)/(16))`
`implies f(x)` has least value `-(9)/(8)`
`f(x) lt 0 implies (2x-5)(x-1) lt 0implies1 lt x lt 5//2`