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Let f(x)=ax^(2)+bx+c, a ne 0, a, b, c in...

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`.
The least value of `f(x)` is

A

`3//4`

B

`9//2`

C

`-9//8`

D

`3//4`

Text Solution

Verified by Experts

The correct Answer is:
C
`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`
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