`{:("Column A" , "A function f(x) is defined for all real numbers as","Column B"),(, f(x)=(x - 1)(x-2)(x-3)(x-4),),( f(2.5), ,f(3.5)):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the problem, we need to evaluate the function \( f(x) = (x - 1)(x - 2)(x - 3)(x - 4) \) at two different points: \( f(2.5) \) and \( f(3.5) \).

### Step 1: Calculate \( f(2.5) \)

1. Substitute \( x = 2.5 \) into the function:
\[ f(2.5) = (2.5 - 1)(2.5 - 2)(2.5 - 3)(2.5 - 4) \]

2. Simplify each term:
- \( 2.5 - 1 = 1.5 \)
- \( 2.5 - 2 = 0.5 \)
- \( 2.5 - 3 = -0.5 \)
- \( 2.5 - 4 = -1.5 \)

3. Now substitute these values back into the function:
\[ f(2.5) = (1.5)(0.5)(-0.5)(-1.5) \]

4. Calculate the product:
- First multiply the positive terms: \( 1.5 \times 0.5 = 0.75 \)
- Then multiply the negative terms: \( -0.5 \times -1.5 = 0.75 \)
- Now multiply these results: \( 0.75 \times 0.75 = 0.5625 \)

Thus, \( f(2.5) = 0.5625 \).

### Step 2: Calculate \( f(3.5) \)

1. Substitute \( x = 3.5 \) into the function:
\[ f(3.5) = (3.5 - 1)(3.5 - 2)(3.5 - 3)(3.5 - 4) \]

2. Simplify each term:
- \( 3.5 - 1 = 2.5 \)
- \( 3.5 - 2 = 1.5 \)
- \( 3.5 - 3 = 0.5 \)
- \( 3.5 - 4 = -0.5 \)

3. Now substitute these values back into the function:
\[ f(3.5) = (2.5)(1.5)(0.5)(-0.5) \]

4. Calculate the product:
- First multiply the positive terms: \( 2.5 \times 1.5 = 3.75 \)
- Then multiply the positive term with the negative term: \( 3.75 \times 0.5 = 1.875 \)
- Finally, multiply by the last negative term: \( 1.875 \times -0.5 = -0.9375 \)

Thus, \( f(3.5) = -0.9375 \).

### Step 3: Compare \( f(2.5) \) and \( f(3.5) \)

Now we have:
- \( f(2.5) = 0.5625 \)
- \( f(3.5) = -0.9375 \)

Since \( 0.5625 > -0.9375 \), we conclude that:

**Column A is larger than Column B.**

### Final Answer:
Column A is larger.

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