[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}