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

Options

A. `(1)/(3)ltIlt(1)/(sqrt8)`
B. `(1)/(4)ltIlt(1)/(3)`
C. `(1)/(4)ltIlt0`
D. none of these

Answer

A

Explanation

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)`