Find the intervals in which the functions are increasing or decreasin

1.	Find the intervals in which the functions are increasing or decreasing. f(x) = x3 – 6x2 – 36x + 2
Answer» Given as f(x) = x³ – 6x² – 36x + 2 f'(x) = (d/dx)(x³ – 6x² – 36x + 2) ⇒ f’(x) = 3x² – 12x – 36 For the function f(x) we have to find critical point, we must have ⇒ f’(x) = 0 ⇒ 3x² – 12x – 36 = 0 ⇒ 3(x² – 4x – 12) = 0 ⇒ 3(x² – 6x + 2x – 12) = 0 ⇒ x² – 6x + 2x – 12 = 0 ⇒ (x – 6) (x + 2) = 0 ⇒ x = 6, – 2 It is clear, f’(x) > 0 if x < –2 and x > 6 and f’(x) < 0 if –2< x < 6 Hence, f(x) increases on (–∞,–2) ∪ (6, ∞) and f(x) is decreasing on interval x ∈ (–2, 6)

Find the intervals in which the functions are increasing or decreasing. f(x) = x3 – 6x2 – 36x + 2

Answer»

Given as f(x) = x³ – 6x² – 36x + 2

f'(x) = (d/dx)(x³ – 6x² – 36x + 2)

⇒ f’(x) = 3x² – 12x – 36

For the function f(x) we have to find critical point, we must have

⇒ f’(x) = 0

⇒ 3x² – 12x – 36 = 0

⇒ 3(x² – 4x – 12) = 0

⇒ 3(x² – 6x + 2x – 12) = 0

⇒ x² – 6x + 2x – 12 = 0

⇒ (x – 6) (x + 2) = 0

⇒ x = 6, – 2

It is clear, f’(x) > 0 if x < –2 and x > 6 and f’(x) < 0 if –2< x < 6

Hence, f(x) increases on (–∞,–2) ∪ (6, ∞) and f(x) is decreasing on interval x ∈ (–2, 6)