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

f(x) = x4 – 4x3 + 4x2 + 15


Given:- Function


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) = 4x3 – 12x2 + 8x


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


⇒ f’(x) = 0


⇒ 4x3 – 12x2 + 8x= 0


⇒ 4(x3 – 3x2 + 2x) = 0


⇒ x(x2 – 3x + 2) = 0


⇒ x(x2 – 2x – x + 2) = 0


⇒ x(x – 2)(x – 1)


⇒ x = 0, 1 , 2





clearly, f’(x) > 0 if 0 < x < 1 and x > 2


and f’(x) < 0 if x < 0 and 1 < x < 2


Thus, f(x) increases on (0, 1) ∪ (2, ∞)


and f(x) is decreasing on interval (–∞, 0) ∪ (1, 2)



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