Find the values of x for which f(x) = [x (x - 2)]² is an increasing function. Also, find the

points on the curve, where the tangent is parallel to x-axis.

Given f(x)= [x(x-2)]²

= x²(x² + 4 – 4x)

= x⁴ + 4x² – 4x³

Now for increasing function,

⇒ 4x³ – 12x² + 8x > 0

⇒ 4x(x – 1)(x – 2) > 0

For 0<x<1,

For x>2,

The function is increasing for 0 < x < 1 and x >2

If tangent is parallel to x- axis then

⇒ 4x(x – 1)(x – 2) = 0

⇒ x = 0,1,2

For x = 0, f(0) = 0

For x = 1, f(1) = [1(1-2)]²

= 1

For x = 2, f(2) = [2× 0]² = 0

∴ points are (0,0),(1,1),(2,0).