Let `f(x) = x2 + b_1x + c_1. g(x) = x^2 + b_2x + c_2`. Real roots of `f(x) = 0` be `alpha, beta` and real roots of `g(x) = 0` be `alpha+gamma, beta+gamma`. Least values of `f(x)` be `- 1/4`Least value of `g(x)` occurs at `x=7/2`

Options

A. `-1`
B. `-1/2`
C. `-1/3`
D. `-1/4`

Answer

D

Explanation

We have `(alpha-beta)=(alpha+k)-(beta+k)`
`implies(sqrt(b^(2)-4c))/1=(sqrt(b_(1)^(2)-4c_(1)))/1`
`impliesb^(2)-4c=b_(1)^(3)-4c_(1)`…i
Given least value of `f(x)=-1/4-((b^(2)-4c))/(4xx1)=-1/4`
`impliesb^(2)-4c=1`
`:.b^(2)-4c=1=b_(1)^(2)-4c_(1)` [from Eq. (i) ]..ii
Also given least value of `g(x)` occurs at `x=7/2`
`:.-(b_(1))/(2xx1)=7/2`
`:.b_(1)=-7`
Least value of `g(x)=-(b_(1)^(2)-4c_(1))/(4xx1)=-1/4` [from Eq. (ii)]