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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

Let `f(x) = ax^(2) + bx + c `
Q `f (1) = f(- 1) rArr a + b +c = a - b + c rArrb = 0 `
Q `f ( x) = ax^(2) + c rArr f' ( x) =2ax `Q `f' (a_(1)) =2aa_(1) , f'( a_(2)) = 2 a a_(2)`
`f' ( a_(3) ) = 2 a a_(3)`
Now assume
`2 f' ( a_(2)) = f' ( a_(1)) + f' ( a_(3))`
`rArr 2.2 a a_(2)= 2 a a_(1) +2 a a_(3) rArr 2a_(2)=a_(1) + a_(3) rArr a_(1), a_(2) , a_(3)` areinAP.
`:. f' ( a_(1)), f'(a_(2)), f'(a_(3)) ` are in AP.
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