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If f(x)=int(2)^(x)(dt)/(1+t^(4)), then...

If `f(x)=int_(2)^(x)(dt)/(1+t^(4))`, then

A

`f(3)lt(1)/(17)`

B

`f(3)gt(1)/(17)`

C

`f(3)=(1)/(17)`

D

`f(3)gt1`

Text Solution

Verified by Experts

The correct Answer is:
A
`f(x)=int_(2)^(x)(dt)/(1+t^(4)).`
`rArr" "f'(x)=(1)/(1+x^(4))`
In [2,3], apply mean value theorem to f(x)
`therefore" "(f(3)-f(2))/(3-2)=f'(x),` where `c in (2,3)`
`therefore" "f(3)-0=(1)/(1+c^(4))`
Now 2 lt c lt 3
`17 lt 1+c^(4)lt84`
`rArr" "(1)/(17)gt(1)/(1+c^(4))gt(1)/(82)`
`rArr" "(1)/(82)ltf(3)lt(1)/(17)`
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