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: Find the first derivative \( f'(x) \)
To find the critical points, we first need to compute the first derivative of the function.

\[ f'(x) = \frac{d}{dx}(x^5 - 5x^4 + 5x^3 - 1) = 5x^4 - 20x^3 + 15x^2 \]
...