Question

Consider the equation below. (If an answer does not exist, enter DNE.) f(x) = x^3 − 3x^2 − 9x + 4 (a) Find the interval on which f is increasing. (Enter your answer using interval notation.)

Answer 1

Answer:

(-∞, -1 )∪(3, ∞)

Step-by-step explanation:

Given function,

$f(x) = x^3 - 3x^2 - 9x + 4$

Differentiating with respect to x,

$f'(x)=3x^2-6x-9$

For increasing or decreasing,

f'(x) = 0

$\implies 3x^2-6x-9=0$

By quadratic formula,

$x=\frac{-(-6)\pm \sqrt{(-6)^2-4\times 3\times -9}}{6}$

$=\frac{ 6\pm \sqrt{36+108}}{6}$

$=\frac{6\pm \sqrt{144}}{6}$

$\frac{6\pm 12}{6}$

$\implies x = 3\text{ or } x = -1$

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, ∞),