Find the values `' a '` for which the function `f(x)=(a+2)x^3-3a x^2+9a x-1` decreases for all real values of `x` .

Explanation

We have
`f(x) = (a + 2)x^(3) - 3ax^(2) + 9ax - 1`
Slope of tangent to the curve at any point on it is
According to the question,
`3(a + 2)x^(2) - 6ax + 9a le 0 for all x in` R.
`therefore a + 2 lt 0 `and discriminant` le 0`
`rArr a lt - 2 and - 8a^(2) - 24 a le 0 `
`rArr a lt - 2 and a (a + 3) ge` 0
`rArr alt - 2 and a le - 3 or a ge 0`
`rArr a le - 3`