Find the points of local maxima or local minima, if any, using first derivative test, and local maximum or local minimum of `f(x)=xsqrt(1-x),\ \ x >0`

Explanation

To find the points of local maxima or local minima for the function \( f(x) = x \sqrt{1 - x} \) for \( x > 0 \), we will use the first derivative test. Here’s a step-by-step solution:

### Step 1: Find the first derivative \( f'(x) \)

Using the product rule, we differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x) \cdot \sqrt{1 - x} + x \cdot \frac{d}{dx}(\sqrt{1 - x}) \]
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