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Find the value of a for which the functi...

Find the value of `a` for which the function `(a+2)x^3-3a x^2+9a x-1` decreases montonically for all real `xdot`

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Verified by Experts

The correct Answer is:
`[-oo,-3]`
If f(x) =`(a+2)x^(3)-3ax^(2)+9ax-1` decreases monotonically for all x in R, then
`f'(x) le 0 for all x in R`
`3(a+2)x^(2)=6ax+9a le 0 for all x in R`
`therfore a+2 le0` and discriminant `le 0`
or `lt -2 and -8a^(2)-24ale0`
`alt -2 and a(a+3)ge0`
`alt -2 and ale -3` or a ge 0`
or `ale -3`
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