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let f(x) be a polynomial function of sec...

let `f(x)` be a polynomial function of second degree. If `f(1)=f(-1)and a_(1),a_(2),a_(3)` are in AP, then show that `f'(a_(1)),f'(a_(2)),f'(a_(3))` are in AP.

A

AP

B

GP

C

HP

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
`"Let "f(x)=ax^(2)+bx+c`
`because" "f(1)=f(-1)" "rArr" "a+b+c=a-b+c rArr b=0`
`because" "f(x)=ax^(2)+c rArr f.(x)=2ax" "because" "f.(x)=2ax" "because" "f.(a_(1))=2aa_(1), f.(a_(2))=2aa_(2)`,
`f.(a_(3))=2aa_(3)`
Now assume
`2f.(a_(2))=f.(a_(1))+f.(a_(3))`
`rArr" "2.2aa_(2)=2aa_(1)+2aa_(3)" "rArr" "2a_(2)=a_(1)+a_(3)" "rArr" "a_(1),a_(2),a_(3)" are in AP"`
`therefore" "f.(a_(1)),f.(a_(2)),f.(a_(3))" are in AP"`.
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