Find the interval to which b may belong so that the function f(x)`=(1-sqrt(21-4b-b^(2))/(b+1)) x^(3) +5x +sqrt(6)` is increasing at every point of its domain.

Options

A. (-7,-1)
B. (-6,-2)
C. (2,25)
D. all of the above

Explanation

For f(x) to be increasing , we must have
f'(x) gt0 for the all x
`rArr 3 (1-sqrt(21-4b-b^2)/(b+1))x^2+5gt0` for all x
`1-(1-sqrt(21-4b-b^2)/(b+1))gt0`
This inequality holds for
b+1 lt 0 and 21-4b-`b^2` gt 0
`rArr` blt-1 and `b^2 -4b-21 lt 0 `
`rArr blt-1" and " -7ltblt3`
`rArr bin (-7,-1)`
Therefore , answer (a) which includes answer (b) also , is correct
Hence , all the options are correct
For b+1gt0 , f(x) will be increasing for all x , if
`21-4b-b^2 gt0 " and " 1-sqrt(21-4b-b^2)/(b+1)gt0`
`rArr b^2+4b-21lt0" and "(b+1)gtsqrt(21-4b-b^2)`
`rArr (b+7) (b-3)lt0 " and " b^2+3b-10gt0`
`rArr -7ltblt3 " and " b lt-5 or b gt 2 `
`rArr b in (2,3)` `[thereforeb+1gt0 thereforeb" gt -1 " ]
Therefore answer (d) , which includes answer (c) also , is correct