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