Question

Let \( f(x) = \int_{2-2x}^{x^2-1} e^{t^2-t} \, dt \) for all \( x \in \mathbb{R} \). If \( f \) is decreasing on \( (0, m) \) and increasing on \( (m, \infty) \), then the value of \( m \) is equal to __________ (answer in integer).

Answer 1

Correct Answer: 1

1. Critical Points of \( f(x) \): - Differentiate \( f(x) \) using the Fundamental Theorem of Calculus: \[ f'(x) = e^{(x^2 - 2x)^2 - (x^2 - 2x)} \cdot \frac{d}{dx}(x^2 - 2x). \] - Compute \( \frac{d}{dx}(x^2 - 2x) \): \[ \frac{d}{dx}(x^2 - 2x) = 2x - 2. \] Thus: \[ f'(x) = e^{(x^2 - 2x)^2 - (x^2 - 2x)} \cdot (2x - 2). \] 2. Find \( m \): - \( f'(x) \) changes sign where \( 2x - 2 = 0 \), i.e., \( x = 1 \). - For \( x \in (0, m) \), \( f'(x) < 0 \) (decreasing), and for \( x \in (m, \infty) \), \( f'(x) > 0 \) (increasing). - Thus, \( m = 1 \)