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### 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)

The function f(x) = -5(x - 2)(x - 4)(x + 1) is given. Determine the coordinates of all local minima and maxima of f(x), or state that there are no local minima or maxima.

Explore how to find the local minima and maxima of the polynomial function f(x) = -5(x - 2)(x - 4)(x + 1) by applying calculus principles, including differentiation and solving critical points using the quadratic formula. Suitable for advanced high school and college students.

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