Question

If \( x = -1 \) and \( x = 2 \) are extreme points of \( f(x) = \alpha \log|x| + \beta x^2 + x \), \( x \neq 0 \), then:

• A. \( \alpha = -6, \beta = \frac{1}{2} \)
• B. \( \alpha = -6, \beta = -\frac{1}{2} \)
• C. \( \alpha = 2, \beta = -\frac{1}{2} \)
• D. \( \alpha = 2, \beta = \frac{1}{2} \)

Hint:

Setting the first derivative to zero at the given extreme points yields a system of equations that can be solved for the unknown parameters.