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Let F:RtoR be a thrice differntiable fun...

Let `F:RtoR` be a thrice differntiable function. Suppose that `F(1)=0,F(3)=-4` and `F'(x)lt0` for all `x epsilon(1//2,3)`. Let `f(x)=xF(x)` for all `x epsilonR`.
The correct statement(s) is (are)

A

`f'(1)lt0`

B

`f(2)lt0`

C

`f'(x)!=0` for an `x epsilon(1,3)`

D

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

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
`f(x)=xF(x)`………….1
`:.f'(x)=xF'(x)+F(x)`…………….2
`impliesf'(1)=F'(1)+F(1)=F'(1)lt0`
`F(1)=0` and `F(3)=-4`
Also `F'(x)gt0` for all `xepsilon(1//2,3)`
So `F(x)` is decreasing and hence `F(2)lt0`.
`:.f(2)-2F(2)lt0`
Also for `x epsilon(1,3)`,
`f'(x)=xF'(x)+F(x)lt0`
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