[THOUGHT] The function (f(x) = 2x^3 + 3x^2 - 12x - 8) must satisfy (-k \leq f(x) \leq k) for all (x \in [-2, 2]). This requires (k) to be at least the maximum of (|f(x)|) over the interval.

First, find the critical points by setting the derivative (f'(x) = 6x^2 + 6x - 12 = 0). Solving (x^2 + x - 2 = 0) gives (x = -2) and (x = 1). These are within ([-2, 2]), along with the endpoints.

Evaluate (f(x)) at (x = -2), (x = 1), and (x = 2):

• (f(-2) = 2(-8) + 3(4) - 12(-2) - 8 = -16 + 12 + 24 - 8 = 12)
• (f(1) = 2(1) + 3(1) - 12(1) - 8 = 2 + 3 - 12 - 8 = -15)
• (f(2) = 2(8) + 3(4) - 12(2) - 8 = 16 + 12 - 24 - 8 = -4)

The absolute values are:

• (|f(-2)| = 12)
• (|f(1)| = 15)
• (|f(2)| = 4)

The maximum absolute value is 15, so (k \geq 15). Testing other points in ([-2, 2]) (e.g., (x = 0), (x = -1), (x = 1.5)) yields values less than 15 in absolute value. Thus, the smallest (k) is 15.

\boxed{15}

[THOUGHT] I will use the Python tool to compute the function values at the critical points and endpoints to find the maximum absolute value. [/THOUGHT] [PYTHON]

Define the function

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

Evaluate at critical points and endpoints

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

\boxed{15}