Find the points of local maxima, local minima and the points of inflection of the function `f(x)=x^5-5x^4+5x^3-1` . Also, find the corresponding local maximum and local minimum values

Explanation

To find the points of local maxima, local minima, and points of inflection for the function \( f(x) = x^5 - 5x^4 + 5x^3 - 1 \), we will follow these steps:

### Step 1: Differentiate the function
We first need to find the first derivative of the function to locate the critical points.

\[ f'(x) = \frac{d}{dx}(x^5 - 5x^4 + 5x^3 - 1) \]
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