Let f(x) = x²+ 3ax + 4, x ∈ [-1, 6] . The range of a. for which f(x) is strictly decreasing function is :

(-∞, -4)

f'(x) = 2x + 3a

∵ f(x) is strictly decreasing

∴ f'(x) < 0 ⇒ 2x + 3a < 0

$⇒ x < \frac{-3a}{2}$

∵ x ∈ [-1, 6]

the maximum value of $a$ for $f$ to be decreasing on $(-1,6)$ is given by, $a_{max} = \frac{-2\times 6}{3}=-4$

∴ Range of a (-∞, -4)