[THOUGHT] We need the smallest positive (k) such that (-k \le f(x) \le k) for all (x \in [-2,2]), where (f(x) = 2x^3 + 3x^2 - 12x - 8). This is exactly [ k = \max_{x \in [-2,2]} |f(x)|. ] We'll use Python (sympy) to:

1. Define (f(x)).
2. Find critical points of (f(x)) in ([-2,2]) via derivative (f'(x)=0).
3. Evaluate (|f(x)|) at those critical points and at the endpoints (-2,2).
4. Take the maximum; that will be the minimal such (k). [/THOUGHT] [PYTHON] import sympy as sp

x = sp.symbols('x', real=True) f = 2x**3 + 3x**2 - 12*x - 8

Derivative and critical points

fprime = sp.diff(f, x) crit_points = sp.solve(sp.Eq(fprime, 0), x) crit_points

Filter critical points to those in [-2,2]

crit_in_interval = [cp for cp in crit_points if cp.is_real and -2 <= float(cp) <= 2]

Points to test: critical points in interval and endpoints -2, 2

test_points = crit_in_interval + [-2, 2]

Evaluate f and |f| at these points

values = [(pt, sp.simplify(f.subs(x, pt)), sp.Abs(sp.simplify(f.subs(x, pt)))) for pt in test_points] values_simplified = [(sp.N(pt), sp.N(val), sp.N(abs_val)) for pt, val, abs_val in values] print("Test points, f(x), |f(x)|:", values_simplified)

Find max |f(x)| among these

max_val = max(abs_val for _, _, abs_val in values_simplified) print("Maximum |f(x)| on [-2,2] is:", max_val)

Also get exact form for that maximum from symbolic data

max_exact = max(values, key=lambda t: t[2])[2] print("Exact maximum |f(x)|:", max_exact) [/PYTHON]

[THOUGHT] The maximum of (|2x^3 + 3x^2 - 12x - 8|) on ([-2,2]) from the Python output is 15, so the smallest (k) satisfying the inequality is 15. [ \boxed{15} ]