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.

Explanation

`y=[x(x-2)]^2=[x^2-2x]`

`dy/dx=y'=2(x^2-2x)(2x-2)=4x(x-2)(x-1)`

`impliesx=0,x=2,x=1`

The points `x=0`, `x=1` and `x=2`
divide the real line into four disjoint intervals, i.e. `(-infty,0),(0,1),(1,2) and (2,infty)`

In intervals `(-infty,0)` and `(1,2)`, `dy/dx<0`

y is strictly decreasing in intervals `(-infty,0)` and `(1,2)`

However in intervals `(0,1)` and `(2,infty)`, `dy/dx>0`