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) = 2x3 + 9x2 + 12x + 20
Answer» Given as f(x) = 2x³ + 9x² + 12x + 20 Differentiate the above equation we get f'(x) = (d/dx)(2x³ + 9x² + 12x + 20) ⇒ f’(x) = 6x² + 18x + 12 For the function f(x) we have to find critical point, we must have ⇒ f’(x) = 0 ⇒ 6x² + 18x + 12 = 0 ⇒ 6(x² + 3x + 2) = 0 ⇒ 6(x² + 2x + x + 2) = 0 ⇒ x² + 2x + x + 2 = 0 ⇒ (x + 2) (x + 1) = 0 ⇒ x = –1, –2 It is clear, f’(x) > 0 if –2 < x < –1 and f’(x) < 0 if x < –1 and x > –2 Hence, f(x) increases on x ∈ (–2,–1) and f(x) is decreasing on interval (–∞, –2) ∪ (–2, ∞)

Find the intervals in which the functions are increasing or decreasing. f(x) = 2x3 + 9x2 + 12x + 20

Answer»

Given as f(x) = 2x³ + 9x² + 12x + 20

Differentiate the above equation we get

f'(x) = (d/dx)(2x³ + 9x² + 12x + 20)

⇒ f’(x) = 6x² + 18x + 12

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

⇒ f’(x) = 0

⇒ 6x² + 18x + 12 = 0

⇒ 6(x² + 3x + 2) = 0

⇒ 6(x² + 2x + x + 2) = 0

⇒ x² + 2x + x + 2 = 0

⇒ (x + 2) (x + 1) = 0

⇒ x = –1, –2

It is clear, f’(x) > 0 if –2 < x < –1 and f’(x) < 0 if x < –1 and x > –2

Hence, f(x) increases on x ∈ (–2,–1) and f(x) is decreasing on interval (–∞, –2) ∪ (–2, ∞)