We are given the function:

$f(x) = x^3 - 1.5x^2 - 6x + 20$

We need to find the values of $x$ where $f'(x) = 0$. Here's how we can proceed analytically:

Step 1: Find $f'(x)$

Differentiate the function with respect to $x$:

$f'(x) = \frac{d}{dx}\left(x^3 - 1.5x^2 - 6x + 20\right)$

Using the power rule for differentiation: $f'(x) = 3x^2 - 3x - 6$

Step 2: Set $f'(x) = 0$

We now set the derivative equal to zero and solve for $x$:

$3x^2 - 3x - 6 = 0$

Divide through by 3 to simplify the equation:

$x^2 - x - 2 = 0$

Step 3: Solve the quadratic equation

Now solve the quadratic equation:

$x^2 - x - 2 = 0$

Factor the quadratic:

$(x - 2)(x + 1) = 0$

This gives the solutions:

$x = 2 \quad \text{or} \quad x = -1$

Final Answer:

The values of $x$ where $f'(x) = 0$ are $x = -1$ and $x = 2$, in increasing order.

Would you like any further details on the steps?

Here are five related questions to expand on this:

1. How do you find the derivative of more complex functions like trigonometric or exponential ones?
2. What is the geometric significance of finding where $f'(x) = 0$?
3. Can every cubic function have exactly two real critical points?
4. How does the power rule for differentiation generalize to polynomials of higher degree?
5. What if the quadratic equation couldn't be factored? How would you solve it then?

Tip: For quadratics that don't factor easily, the quadratic formula $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ is a reliable method to find the roots.