To determine the intervals in which the function $f(x) = (x-1)(x-2)(x-3)$ is negative, we need to analyze the sign of $f(x)$ in different intervals. We start by identifying the roots of the function, which are $x = 1$, $x = 2$, and $x = 3$.

These roots divide the real number line into four intervals:

1. $x < 1$
2. $1 < x < 2$
3. $2 < x < 3$
4. $x > 3$

We will test the sign of $f(x)$ in each of these intervals.

1. Interval $x < 1$

Choose a test point $x = 0$: $f(0) = (0-1)(0-2)(0-3) = (-1)(-2)(-3) = -6$ Since $-6$ is negative, $f(x)$ is negative in this interval.

2. Interval $1 < x < 2$

Choose a test point $x = 1.5$: $f(1.5) = (1.5-1)(1.5-2)(1.5-3) = (0.5)(-0.5)(-1.5) = 0.375$ Since $0.375$ is positive, $f(x)$ is positive in this interval.

3. Interval $2 < x < 3$

Choose a test point $x = 2.5$: $f(2.5) = (2.5-1)(2.5-2)(2.5-3) = (1.5)(0.5)(-0.5) = -0.375$ Since $-0.375$ is negative, $f(x)$ is negative in this interval.

4. Interval $x > 3$

Choose a test point $x = 4$: $f(4) = (4-1)(4-2)(4-3) = (3)(2)(1) = 6$ Since $6$ is positive, $f(x)$ is positive in this interval.

Based on this analysis, $f(x) = (x-1)(x-2)(x-3)$ is negative in the intervals $(-\infty, 1)$ and $(2, 3)$.

Would you like more details or have any questions?

Here are 5 questions you might want to ask:

1. How do we find the intervals where a polynomial function is positive or negative?
2. Can you explain the process of testing intervals in more detail?
3. How does the sign of a function change at its roots?
4. What are some example problems involving polynomial functions?
5. How do we graph polynomial functions and their intervals?

Tip: When analyzing the sign of a polynomial function, evaluate the function at points within each interval determined by its roots. This helps to identify where the function is positive or negative.