Question

Let `f(x)=ax^(2)+bx+c`, `g(x)=ax^(2)+qx+r`, where `a`, `b`, `c`,`q`, `r in R` and `a lt 0`. If `alpha`, `beta` are the roots of `f(x)=0` and `alpha+delta`, `beta+delta` are the roots of `g(x)=0`, then
A. `f_(max) gt g_(max)`
B. `f_(max) lt g_(max)`
C. `f_(max) = g_(max)`
D. cant say anything about relation between `f_(max)` and `g_(max)`

Answer 1

Correct Answer - C
`(c )` `|alpha-beta|^(2)=|(alpha+delta)-(beta+delta)|^(2)`
`implies(alpha+beta)^(2)-4alphabeta`
`=((alpha+delta)+(beta+delta))^(2)-4(alpha+delta)(beta+delta)`
`implies (b^(2))/(a^(2))-(4c)/(a)=(q^(2))/(a^(2))-(4r)/(a)`
`implies (b^(2)-4ac)/(4a)=(q^(2)-4ar)/(4a)`
`impliesf_(max)=-(D)/(4a)=g_(max)`