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The function given is \( f(x) = (x^2 - 2x - 8)(x - 1) \), and the goal is to find the range of values of \( x \) for which \( f(x) \geq 0 \) given that \( 0 \leq x \leq \frac{3}{2} \).

### Step 1: Factorize the quadratic expression
First, we factor the quadratic expression \( x^2 - 2x - 8 \).

We can factor it as:
\[
x^2 - 2x - 8 = (x - 4)(x + 2)
\]

So, the function becomes:
\[
f(x) = (x - 4)(x + 2)(x - 1)
\]

### Step 2: Find the roots of the function
The function will be zero when any of the factors are zero. The roots are:
- \( x - 4 = 0 \Rightarrow x = 4 \)
- \( x + 2 = 0 \Rightarrow x = -2 \)
- \( x - 1 = 0 \Rightarrow x = 1 \)

These roots divide the real line into different intervals. We are interested in the behavior of the function within the interval \( [0, \frac{3}{2}] \).

### Step 3: Analyze the sign of the function in \( [0, \frac{3}{2}] \)
We now check the sign of \( f(x) \) within the given interval:

- At \( x = 0 \), \( f(0) = (0 - 4)(0 + 2)(0 - 1) = (-4)(2)(-1) = 8 \) (positive).
- At \( x = 1 \), \( f(1) = (1 - 4)(1 + 2)(1 - 1) = (-3)(3)(0) = 0 \).
- At \( x = \frac{3}{2} \), \( f\left(\frac{3}{2}\right) = \left(\frac{3}{2} - 4\right)\left(\frac{3}{2} + 2\right)\left(\frac{3}{2} - 1\right) \).

Let's calculate \( f\left(\frac{3}{2}\right) \):

\[
f\left(\frac{3}{2}\right) = \left(-\frac{5}{2}\right)\left(\frac{7}{2}\right)\left(\frac{1}{2}\right) = \left(-\frac{35}{8}\right)
\]
which is negative.

### Step 4: Conclusion
The function \( f(x) \geq 0 \) between \( 0 \) and \( 1 \). For values of \( x \) in \( \left( 1, \frac{3}{2} \right] \), the function becomes negative.

Thus, the range of values of \( x \) for which \( f(x) \geq 0 \) in the interval \( 0 \leq x \leq \frac{3}{2} \) is:
\[
x \in [0, 1]
\]

Would you like further details on any step? Here are five related questions to expand on this:

1. How can you verify the behavior of \( f(x) \) for other intervals outside \( [0, \frac{3}{2}] \)?
2. What role do critical points play in finding the sign of a function?
3. How do you determine the concavity of a function like \( f(x) \)?
4. Can you extend this problem to find where \( f(x) > 0 \)?
5. How would the analysis change if the interval for \( x \) were different?

**Tip:** When solving inequalities involving factored polynomials, it is useful to evaluate the sign of each factor across the intervals formed by the roots.

f(x) = (x^2 - 2x - 8)(x - 1). If 0 ≤ x ≤ 3/2, then find the range of values of x for which f(x) ≥ 0.

Explore how to solve the inequality f(x) = (x^2 - 2x - 8)(x - 1) ≥ 0 within the interval 0 ≤ x ≤ 3/2. This problem involves quadratic factorization, polynomial roots, and sign analysis. Ideal for students in Grades 9-11.

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