[THOUGHT] To find the smallest possible 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]), I need to determine the maximum absolute value of the function ( f(x) = 2x^3 + 3x^2 - 12x - 8 ) over this interval. The maximum value of ( |f(x)| ) will give me the smallest ( k ) that satisfies the given inequality.

First, I will find the critical points of ( f(x) ) by taking its derivative and setting it to zero. Then, I will evaluate ( f(x) ) at the critical points and at the endpoints of the interval ([-2, 2]). Finally, I will determine the maximum absolute value of these evaluations. [/THOUGHT] [PYTHON] from sympy import symbols, diff, solve

Define the variable and the function

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

Calculate the derivative

f_prime = diff(f, x)

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

critical_points = solve(f_prime, x)

Evaluate the function at the critical points and the endpoints of the interval [-2, 2]

endpoints = [-2, 2] all_points = critical_points + endpoints all_points = [point.evalf() for point in all_points if -2 <= point <= 2] # Filter points in the interval

Evaluate f(x) at these points

values = [f.subs(x, point).evalf() for point in all_points]

Find the maximum absolute value of these evaluations

k = max(abs(value) for value in values) print(k)

[THOUGHT] Upon reviewing the Python code execution, I realize there was an error in handling the data types. I will re-evaluate the critical points and endpoints manually to ensure accuracy.

The derivative ( f'(x) = 6x^2 + 6x - 12 ). Solving ( 6x^2 + 6x - 12 = 0 ) gives ( x^2 + x - 2 = 0 ), which factors to ( (x+2)(x-1) = 0 ), yielding critical points ( x = -2 ) and ( x = 1 ).

Evaluating ( f(x) ) at the critical points and endpoints:

• ( 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 maximum absolute value among ( 12, -15, -4 ) is ( 15 ). Therefore, the smallest possible positive value of ( k ) is ( 15 ). [/THOUGHT] \boxed{15}