Step-by-Step Solution for Finding Local Minima and Maxima of $f(x)$

We are given the polynomial function $f(x) = -5(x - 2)(x - 4)(x + 1)$ and asked to determine the coordinates of any local minima and maxima. Here's the process:

1. Find the First Derivative $f'(x)$:
   To locate critical points, we first compute the derivative of $f(x)$. Using the product rule and chain rule:

   $f(x) = -5(x - 2)(x - 4)(x + 1)$

   We'll first expand this product:

   $f(x) = -5[(x - 2)(x - 4)](x + 1)$ First simplify $(x - 2)(x - 4)$: $(x - 2)(x - 4) = x^2 - 6x + 8$ So, $f(x) = -5(x^2 - 6x + 8)(x + 1)$ Now distribute $(x + 1)$ across $(x^2 - 6x + 8)$: $(x^2 - 6x + 8)(x + 1) = x^3 - 5x^2 + 2x + 8$ Hence, $f(x) = -5(x^3 - 5x^2 + 2x + 8)$ Expanding this gives: $f(x) = -5x^3 + 25x^2 - 10x - 40$

2. Differentiate $f(x)$ to Find $f'(x)$: Now we differentiate $f(x) = -5x^3 + 25x^2 - 10x - 40$: $f'(x) = -15x^2 + 50x - 10$

3. Find the Critical Points: To find the critical points, set $f'(x) = 0$: $-15x^2 + 50x - 10 = 0$ We can divide the entire equation by -5 to simplify: $3x^2 - 10x + 2 = 0$ Solve this quadratic using the quadratic formula: [ x = \frac{-(-10)