Question

Let \( f(x) = \int_0^x e^t (t - 1)(t - 2) \, dt \). Then \( f(x) \) decreases in the interval

• A. ( x \in (1, 2) \)
• B. ( x \in (2, 3) \)
• C. ( x \in (0, 1) \)
• D. ( x \in (0.5, 1) \)

Hint:

To determine where a function is increasing or decreasing, check the sign of its derivative. A positive derivative indicates increasing behavior, while a negative derivative indicates decreasing behavior.

Answer 1

The Correct Option is A

Step 1: Differentiating \( f(x) \).
The function \( f(x) \) is defined as an integral. To find where \( f(x) \) decreases, we first compute its derivative with respect to \( x \) using the Fundamental Theorem of Calculus: \[ f'(x) = e^x (x - 1)(x - 2). \] Step 2: Analyzing \( f'(x) \).
The expression \( f'(x) = e^x (x - 1)(x - 2) \) shows that the function’s sign depends on the factors \( (x - 1) \) and \( (x - 2) \). - For \( x \in (1, 2) \), both \( (x - 1) \) and \( (x - 2) \) are negative, so \( f'(x) \) is positive, and \( f(x) \) increases. - Therefore, \( f(x) \) decreases for \( x \in (1, 2) \), which makes (A) the correct answer.
Step 3: Conclusion.
The correct answer is (A), as this interval corresponds to when \( f(x) \) is decreasing.