[THOUGHT] I need to find the smallest positive (k) such that (|2x^3 + 3x^2 - 12x - 8| \le k) for all (x \in [-2,2]). This means (k) must be the maximum absolute value of the cubic function (f(x)=2x^3+3x^2-12x-8) on the interval. I will:

1. Compute the derivative (f'(x)) and find its critical points inside ([-2,2]).
2. Evaluate (f(x)) at the critical points and at the endpoints (-2) and (2).
3. Take the absolute values of these evaluations and determine the largest one.
4. Verify by evaluating (f(x)) on a fine grid of points in the interval to ensure no larger absolute value is missed.

I will use SymPy for symbolic computation and a numeric check. [/THOUGHT] [PYTHON] import sympy as sp

Define variable and function

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

Find critical points by solving f'(x) = 0

f_prime = sp.diff(f, x) critical_points = sp.solve(f_prime, x)

Filter critical points that lie within the interval [-2, 2]

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

Include endpoints

evaluation_points = [-2, 2] + valid_critical

Evaluate f at these points and compute absolute values

abs_values = [abs(f.subs(x, pt).evalf()) for pt in evaluation_points]

Find the maximum absolute value

max_abs = max(abs_values)

Numeric verification on a fine grid

import numpy as np grid = np.linspace(-2, 2, 100000) grid_vals = np.abs(np.array([f.subs(x, float(g)).evalf() for g in grid])) grid_max = np.max(grid_vals)

Print results

print("Critical points within interval:", valid_critical) print("Function values at evaluation points:", [f.subs(x, pt).evalf() for pt in evaluation_points]) print("Absolute values:", abs_values) print("Maximum absolute value from critical points/endpoints:", max_abs) print("Maximum absolute value from fine grid:", grid_max) [/PYTHON]

[THOUGHT] The Python output confirms that the maximum absolute value of the function (2x^3 + 3x^2 - 12x - 8) on the interval ([-2, 2]) is approximately 15. The critical points and endpoints yield values with absolute values 12, 4, and 15, and the fine grid verification supports this result. Thus, the smallest (k) satisfying the inequality for all (x \in [-2, 2]) is 15. [/THOUGHT] \boxed{15}