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

f(x) = (x – 1) (x – 2)^{2}

Given:- Function f(x) = (x – 1) (x – 2)^{2}

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) = (x – 1) (x – 2)^{2}

⇒

⇒ f’(x) =(x – 2)^{2} +2(x – 2)(x – 1)

⇒ f’(x) = (x – 2)(x – 2 + 2x – 2)

⇒ f’(x) = (x – 2 )(3x – 4)

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

⇒ f’(x) = 0

⇒ (x – 2 )(3x – 4) = 0

⇒ x = 2,

clearly, f’(x) > 0 if and x >2

and f’(x) < 0 if

Thus, f(x) increases on

and f(x) is decreasing on interval

1