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Let F : R to R be a thrice differentia...

Let F : R `to` R be a thrice differentiable function . Suppose that `F(1)=0,F(3)=-4 and F(x) lt 0 " for all" x in (1/2,3).` f(x) = x F(x) for all `x inR`.
The correct statement (s) is / are

A

`f'(1)lt0`

B

`f(2)lt0`

C

`f'(x)ne0 " for any" in(1,3)`

D

` f' (x)=0 " for some "x in(1,3)`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
According to the given data , `F'(x)lt0,AA x in (1,3)`
We have , ` f(x) =xF(x)`
`rArrf'(x)=F(x)+xF'(x)` … (i)
`rArrf'(1)=F(1)+F'(1)lt0`
given F (1) = 0 and F ' ` (x) lt 0]`
Also , `f(2)=2F(2)lt0` " " [usingF (x) `lt 0, AA x in (1,3)]`
Now , f ' (x) = F (x) +x F ' 9x) `lt 0` " " [usingF (x) `lt 0, AA x in (1,3)]`
`rArr f' (x) lt0`
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