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Let I=int1^2 (dx)/sqrt(2x^3-9x^2+12x+4)...

Let `I=int_1^2 (dx)/sqrt(2x^3-9x^2+12x+4)` then

A

`(1)/(3)ltIlt(1)/(sqrt8)`

B

`(1)/(4)ltIlt(1)/(3)`

C

`(1)/(4)ltIlt0`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
Let `f(x)=2x^(3)-9x^(2)+12x+4.`
`rArr" "f'(x)=6x^(2)-18x+12=6(x-1)(x-2)le 0 " for " x in [1,2].`
`rArr" "f(x)" is decreasing in "[1,2]`
`therefore" "f(2) lt f(2) lt f(1)`
`therefore" "8lt f(x) lt 9`
`therefore" "(1)/(3)lt(1)/(sqrt(2x^(3)-9x^(2)+12x+4))lt(1)/(sqrt8)`
`rArr" "(1)/(3)int_(1)^(2)dxltint_(1)^(2)(dx)/(sqrt(2x^(3)-9x^(2)+12x+4))lt(1)/(sqrt8)int_(1)^(2)dx`
`rArr" "(1)/(3)ltIlt(1)/(sqrt8)`
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