Find the values of x for which y = [x(x – 2)]² is an increasing function.

It is given that y = [x(x – 2)]², then,

= 4x(x - 2)(x - 1)

Now if

⇒ x = 0, 1,2

So, the points x = 0, x =1 and x = 2 divides the real line into four disjoint intervals,

(-∞,0), (0,1), (1, 2) and (2,∞).

So, in interval(-∞,0),(1,2)

< 0

Therefore, the given function (f) is strictly decreasing in intervals .

So, in interval (0,1) and (2,∞)

Therefore, the given function (f) is strictly increasing for 0 < x < 1 and x>2