Q. 1 X5.0( 2 Votes )
Find the intervals in which the following functions are increasing or decreasing.
f(x) = x3 – 6x2 + 9x + 15
Answer :
Given:- Function f(x) = x3 – 6x2 + 9x + 15
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) = x3 – 6x2 + 9x + 15
⇒ 
⇒ f’(x) = 3x2 – 12x + 9
For f(x) lets find critical point, we must have
⇒ f’(x) = 0
⇒ 3x2 – 12x + 9 = 0
⇒ 3(x2 – 4x + 3) = 0
⇒ 3(x2 – 3x – x + 3) = 0
⇒ x2 – 3x – x + 3 = 0
⇒ (x – 3)(x – 1) = 0
⇒ x = 1, 3
clearly, f’(x) > 0 if x < 1 and x > 3
and f’(x) < 0 if 1 < x < 3
Thus, f(x) increases on (–∞, 1) ∪ (3, ∞)
and f(x) is decreasing on interval x ∈ (1, 3)
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