the interval in which f(x) = (x-1) * (x-2) * (x-3) is negative os

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.

Explore how to determine intervals where the function f(x) = (x-1)(x-2)(x-3) is negative with detailed interval testing. This problem covers polynomial functions and interval analysis, essential for advanced high school students.

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