[THOUGHT] Evaluate $f(x) = 2x^3 + 3x^2 - 12x - 8$ at critical points $x = -2, 1, 2$ to find the smallest $k$. Calculate values and compare absolute magnitudes. [/THOUGHT] [PYTHON] def f(x): return 2 * x3 + 3 * x2 - 12 * x - 8

Evaluate the function at the critical points and endpoints

critical_points = [-2, 1, 2] results = [f(p) for p in critical_points]

max_abs_val = max(abs(r) for r in results) max_abs_val [/PYTHON] $\boxed{15}$[THOUGHT] Evaluate $f(x) = 2x^3 + 3x^2 - 12x - 8$ at critical points $x = -2, 1, 2$ to find the smallest $k$. Calculate values and compare absolute magnitudes. [/THOUGHT] [PYTHON] def f(x): return 2 * x3 + 3 * x2 - 12 * x - 8

Evaluate the function at the critical points and endpoints

critical_points = [-2, 1, 2] results = [f(p) for p in critical_points]

max_abs_val = max(abs(r) for r in results) max_abs_val [/PYTHON] $\boxed{15}$

[THOUGHT] The minimum $k$ occurs at $x=1$ yielding $f(1) = -15$, thus $k \geq 15$. [/THOUGHT] $\boxed{15}$[THOUGHT] The minimum $k$ occurs at $x=1$ yielding $f(1) = -15$, thus $k \geq 15$. [/THOUGHT] $\boxed{15}$