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) = 8 + 36x + 3x2 – 2x3
Answer» Given as f(x) = 8 + 36x + 3x² – 2x³ Differentiate with respect to x f'(x) = (d/dx)(8 + 36x + 3x² - 2x³) ⇒ f’(x) = 36 + 6x – 6x² For the f(x) we have to find critical point, we must have ⇒ f’(x) = 0 ⇒ 36 + 6x – 6x² = 0 ⇒ 6(–x² + x + 6) = 0 ⇒ 6(–x² + 3x – 2x + 6) = 0 ⇒ –x² + 3x – 2x + 6 = 0 ⇒ x² – 3x + 2x – 6 = 0 ⇒ (x – 3) (x + 2) = 0 ⇒ x = 3, – 2 It is clear, f’(x) > 0 if –2 < x < 3 and f’(x) < 0 if x < –2 and x > 3 Hence, f(x) increases on x ∈ (–2, 3) and f(x) is decreasing on interval (–∞, 2) ∪ (3, ∞)

Find the intervals in which the functions are increasing or decreasing. f(x) = 8 + 36x + 3x2 – 2x3

Answer»

Given as f(x) = 8 + 36x + 3x² – 2x³

Differentiate with respect to x

f'(x) = (d/dx)(8 + 36x + 3x² - 2x³)

⇒ f’(x) = 36 + 6x – 6x²

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

⇒ f’(x) = 0

⇒ 36 + 6x – 6x² = 0

⇒ 6(–x² + x + 6) = 0

⇒ 6(–x² + 3x – 2x + 6) = 0

⇒ –x² + 3x – 2x + 6 = 0

⇒ x² – 3x + 2x – 6 = 0

⇒ (x – 3) (x + 2) = 0

⇒ x = 3, – 2

It is clear, f’(x) > 0 if –2 < x < 3 and f’(x) < 0 if x < –2 and x > 3

Hence, f(x) increases on x ∈ (–2, 3) and f(x) is decreasing on interval (–∞, 2) ∪ (3, ∞)