Let `f'(x)=(192x^(3))/(2+sin^(4)pix)` for all `x in R` with `f((1)/(2))=0`. If `m le int_((1)/(2))^(1)f(x)dx leM`, then the possible values of m and M are-

Options

A. m=13,M=24
B. `m=(1)/(4), M=(1)/(2)`
C. m=-11, M=0
D. m=1,M=12

Answer

D

Explanation

We have,
`f'(x)=(192x^(3))/(2+sin^(4)pix)` for all ` x in R`
`rArr (192)/(3)x^(3)lef'(x)dxle(192)/(2)x^(3)` for all `x in [(1)/(2),1]`
`rArr 64 overset(x)underset(1//2)int x^(3)dx le overset(x)underset(1//2)int f'(x)dx le 96 overset(x)underset(1//2)int x^(3)dx` for all `x in ((1)/(2),1)`
`rArr 16x^(4)-1le f(x)-f((1)/(2)) le 24x^(4)-(3)/(2)` for all `x in ((1)/(2),1)`
`rArr 16x^(4)-1 le f(x) le 24x^(4)-(3)/(2)` for all `x in ((1)/(2),1)`
`rArr overset(1)underset(1//2)int (16x^(2)-1)dx le overset(1)underset(1//2)int f(x)dx le overset(1)underset(1//2)int (24x^(4)-(3)/(2))dx`
`rArr 2.6 le overset(1)underset(1//2)int f(x)dx le 3.9`
Clearly, option (d)is correct.