Answer:
(-∞, -1 )∪(3, ∞)
Step-by-step explanation:
Given function,
[tex]f(x) = x^3 - 3x^2 - 9x + 4[/tex]
Differentiating with respect to x,
[tex]f'(x)=3x^2-6x-9[/tex]
For increasing or decreasing,
f'(x) = 0
[tex]\implies 3x^2-6x-9=0[/tex]
By quadratic formula,
[tex]x=\frac{-(-6)\pm \sqrt{(-6)^2-4\times 3\times -9}}{6}[/tex]
[tex]=\frac{ 6\pm \sqrt{36+108}}{6}[/tex]
[tex]=\frac{6\pm \sqrt{144}}{6}[/tex]
[tex]\frac{6\pm 12}{6}[/tex]
[tex]\implies x = 3\text{ or } x = -1[/tex]
In interval (-∞, -1 ), f'(x) = positive,
⇒ f(x) is increasing on (-∞, -1 ),
In interval (-1, 3), f'(x) = negative,
⇒ f(x) is decreasing on (-1, 3),
In interval (3, ∞), f'(x) = positive,
⇒ f(x) is increasing on (3, ∞),
