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Find the intervals in which the following functions are increasing or decreasing.
f(x) = 2x3 – 9x2 + 12x – 5
Given:- Function f(x) = 2x3 – 9x2 + 12x – 5
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 – 5
⇒
⇒ 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 x < 1 and x > 2
and f’(x) < 0 if 1 < x < 2
Thus, f(x) increases on (–∞, 1) ∪ (2, ∞)
and f(x) is decreasing on interval x ∈ (1,2)