Find the points at which the function f given by `f(x)=(x-2)^4(x+1)^3`has
(i) local maxima
(ii) local minima
(iii) point of inflexion

Explanation

To find the points at which the function \( f(x) = (x-2)^4 (x+1)^3 \) has local maxima, local minima, and points of inflection, we will follow these steps:

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

Using the product rule, we differentiate \( f(x) \):

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