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

f(x) = x3 – 6x2 + 9x + 15

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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