Find the intervals in which the following functions are increasing or decreasing.

f(x) = 2x^{3} + 9x^{2} + 12x + 20

Given:- Function f(x) = 2x^{3} + 9x^{2} + 12x + 20

Theorem:- Let f be a differentiable real function defined on an open interval (a,b).

(i) If f’(x) > 0 for all , then f(x) is increasing on (a, b)

(ii) If f’(x) < 0 for all , then f(x) is decreasing on (a, b)

Algorithm:-

(i) Obtain the function and put it equal to f(x)

(ii) Find f’(x)

(iii) Put f’(x) > 0 and solve this inequation.

For the value of x obtained in (ii) f(x) is increasing and for remaining points in its domain it is decreasing.

Here we have,

f(x) = 2x^{3} + 9x^{2} + 12x + 20

⇒

⇒ f’(x) = 6x^{2} + 18x + 12

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

⇒ f’(x) = 0

⇒ 6x^{2} + 18x + 12 = 0

⇒ 6(x^{2} + 3x + 2) = 0

⇒ 6(x^{2} + 2x + x + 2) = 0

⇒ x^{2} + 2x + x + 2 = 0

⇒ (x + 2)(x + 1) = 0

⇒ x = –1, –2

clearly, f’(x) > 0 if –2 < x < –1

and f’(x) < 0 if x < –1 and x > –2

Thus, f(x) increases on x ∈ (–2,–1)

and f(x) is decreasing on interval (–∞, –2) ∪ (–2, ∞)

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