Questions: Find all open intervals on which f(x) = x / (x^2 + x - 2) is decreasing. a. (-∞, ∞) b. (-∞, 0) c. (-∞,-2) ∪ (1, ∞) d. (-∞,-2) ∪ (-2,1) ∪ (1, ∞) e. none of these

Solution Steps

We start with the function \( f(x) = \frac{x}{x^2 + x - 2} \).

Using the quotient rule, we find the derivative \( f'(x) \): \[ f'(x) = \frac{(x^2 + x - 2)(1) - x(2x + 1)}{(x^2 + x - 2)^2} \] This simplifies to: \[ f'(x) = \frac{x^2 - x(2x + 1) + x - 2}{(x^2 + x - 2)^2} \]

To find where \( f(x) \) is decreasing, we need to determine where \( f'(x) < 0 \). The critical points occur at: \[ (-\infty < x < -2) \quad \text{or} \quad (-2 < x < 1) \quad \text{or} \quad (1 < x < \infty) \] Thus, the function \( f(x) \) is decreasing on the intervals \( (-\infty, -2) \), \( (-2, 1) \), and \( (1, \infty) \).

Final Answer