Find the intervals in which `f(x)=(x-1)(x-2)^2` is increasing or decreasing.

Explanation

To find the intervals in which the function \( f(x) = (x-1)(x-2)^2 \) is increasing or decreasing, we will follow these steps:

### Step 1: Find the derivative of the function
We start by applying the product rule to differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}[(x-1)(x-2)^2] \]
Using the product rule, where \( u = (x-1) \) and \( v = (x-2)^2 \):
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