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Find the intervals in which the following functions are increasing or decreasing.
f(x) = 2x3 + 9x2 + 12x + 20
Given:- Function f(x) = 2x3 + 9x2 + 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) = 2x3 + 9x2 + 12x + 20
⇒
⇒ f’(x) = 6x2 + 18x + 12
For f(x) lets 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
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, ∞)