If f(x) = alog_e|x| + bx^2 + x has critical points at x = -2 and x =…

The correct answer is Option (3) → \(a + 4b = 0\)

At a critical point, \(f'(x) = 0\). Since \( \displaystyle\frac{d}{dx}\log_e|x| = \frac{1}{x} \) for \(x \ne 0\):

    \( \displaystyle f'(x) = \frac{a}{x} + 2bx + 1 \)

At \(x = 1\):

    \(a + 2b + 1 = 0\)
    \(a + 2b = -1 \quad \text{...(1)}\)

At \(x = -2\):

    \( \displaystyle -\frac{a}{2} - 4b + 1 = 0 \)

    \(a + 8b = 2 \quad \text{...(2)}\)

Subtracting (1) from (2):

    \(6b = 3\)

    \( \displaystyle b = \frac{1}{2} \)

    \(a = -1 - 2b = -2\)

Checking the options:

    \(a + 4b = -2 + 2 = 0\)

For the other options, \(a + 2b = -1\), \(a - 2b = -3\) and \(a - 4b = -4\), none of which is 0.