Question

Find the value of x for which the function `f(x) = x^3 - 3x^2 - 9x + 25` is increasing.

Answer 1

`f(x) = x^3 - 3x^2 - 9x + 25`
Diffrentiating w.r.t.x
`f(x) = 3x^2 - 6x - 9`
If f is increasing then f(x) > 0
`3x^2 - 6x - 9 > 0`   
`3(x^2 - 2x -  3) > 0`
`3 (x^2 - 3x + x - 3) > 0`
`3[x(x -3) + (x - 3)] > 0`
`3[(x - 3) (x + 1)] > 0`

x - 3 > 0        and x + 1 > 0
or (x - 3) < 0   and (x + 1) < 0
For x - 3 > 0 ⇒ x > 3
and for x + 1 > 0 ⇒ x > -1
For x - 3 < 0 ⇒ x < 3
and for x + 1 < 0 ⇒  x < -1
(x - 3) ( x+ 1) > 0 for x < -1

Function is increasing for x ∈ (3, ∞) 
or for x ∈ ( -∞, -1)