Question

Factor theorem ...first part​

Answer 1

Answer:

(-∞, -2) and (2, ∞)

Step-by-step explanation:

ƒ(x) = x³ -12x + 10

f'(x) = 3x² - 12  

Set 3x² - 12 = 0

        x² -  4 = 0

              x² = 4

               x = ±2

The points x = -2 and x = + 2 divide the number line into three intervals:

(-∞, -2), (-2, +2), and (+2, ∞).

a. Interval (-∞, -2)

x < -2, so f'(x) > 0 when -∞ < x < -2

The function is increasing in (-∞, -2).

b. Interval (-2, 2)

|x| < 2, so f'(x) <0

The function is decreasing in (-2, 2).

c. Interval (2, ∞)

x > 2, so f'(x) > 0 when 2 < x < ∞

The function is increasing in (2, ∞).

Thus, the function is increasing in the intervals (-∞, -2) and (2, ∞).