[THOUGHT] To find the smallest positive value of ( k ) such that the inequality ( -k \leq 2x^3 + 3x^2 - 12x - 8 \leq k ) holds for all ( x ) in the interval ([-2, 2]), we need to determine the maximum absolute value of the function ( f(x) = 2x^3 + 3x^2 - 12x - 8 ) on this interval. This involves evaluating ( f(x) ) at the critical points and the endpoints of the interval.

First, we find the critical points by setting the derivative ( f'(x) ) to zero: [ f'(x) = 6x^2 + 6x - 12 = 0 ] Solving this quadratic equation, we get: [ x^2 + x - 2 = 0 \implies (x + 2)(x - 1) = 0 \implies x = -2 \text{ or } x = 1 ] The critical points are ( x = -2 ) and ( x = 1 ). Since ( x = -2 ) is an endpoint, we only need to consider ( x = 1 ) as an internal critical point. We evaluate ( f(x) ) at the endpoints ( x = -2 ), ( x = 2 ), and the critical point ( x = 1 ).

Next, we compute the function values at these points: [ f(-2) = 2(-2)^3 + 3(-2)^2 - 12(-2) - 8 = -16 + 12 + 24 - 8 = 12 ] [ f(1) = 2(1)^3 + 3(1)^2 - 12(1) - 8 = 2 + 3 - 12 - 8 = -15 ] [ f(2) = 2(2)^3 + 3(2)^2 - 12(2) - 8 = 16 + 12 - 24 - 8 = -4 ]

The absolute values of these function values are: [ |f(-2)| = 12, \quad |f(1)| = 15, \quad |f(2)| = 4 ] The maximum absolute value is 15, which occurs at ( x = 1 ). Therefore, the smallest possible positive value of ( k ) is 15.

To confirm, we can use Python to evaluate the function at these points and verify the maximum absolute value. [/THOUGHT]

[PYTHON] def f(x): return 2x**3 + 3x**2 - 12*x - 8

Evaluate the function at the critical points and endpoints

points = [-2, 1, 2] values = [f(x) for x in points] abs_values = [abs(v) for v in values] max_abs = max(abs_values) print(max_abs) [/PYTHON]

[THOUGHT] The Python output confirms that the maximum absolute value of the function on the interval ([-2, 2]) is 15. Therefore, the smallest possible positive value of ( k ) is 15. [/THOUGHT] \boxed{15}