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(A) function $f(x)$ is monotonic if it is either non-increasing or non-decreasing for all $x \in \mathbb{R}$.This implies that its derivative $f'(x)$

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$(-6, 6)$

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(A) function $f(x)$ is monotonic if it is either non-increasing or non-decreasing for all $x \in \mathbb{R}$.This implies that its derivative $f'(x)$ must not change sign.Given $f(x) = 4x^3 + ax^2 + 3x - 2$, we find the derivative:$f'(x) = 12x^2 + 2ax + 3$.For $f(x)$ to be monotonic, $f'(x) \geq 0$ or $f'(x) \leq 0$ for all $x$.Since the coefficient of $x^2$ is $12 > 0$, the parabola $f'(x)$ opens upwards, so we must have $f'(x) \geq 0$ for all $x$.This condition holds if the discriminant $D \leq 0$.The discriminant $D = (2a)^2 - 4(12)(3) = 4a^2 - 144$.Setting $D \leq 0$ gives $4a^2 - 144 \leq 0$, which simplifies to $a^2 \leq 36$.Thus, $-6 \leq a \leq 6$, or $a \in [-6, 6]$.Comparing this with the given options, the interval $(-6, 6)$ is the most appropriate subset provided.

For a real number $a$, if a real-valued function $f(x) = 4x^3 + ax^2 + 3x - 2$ is monotonic in its domain, then the range of $a$ is

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