Problem Analysis #31

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Problem Statement

For all real numbers xx with −2≤x≤2-2 \leq x \leq 2, the inequality−k≤2x3+3x2−12x−8≤k-k \leq 2x^3 + 3x^2 - 12x - 8 \leq kholds. Find the smallest possible positive value of kk. [3 points]\

Ground Truth
15
Execution Trace
Reasoning Process

To solve the problem, we are given the inequality:

−k≤2x3+3x2−12x−8≤k-k \leq 2x^3 + 3x^2 - 12x - 8 \leq k

for all x∈[−2,2]x \in [-2, 2]. Our goal is to find the smallest positive value of kk such this is true for all xx in the interval.

Step 1: Understand the Problem

We are given a function:

f(x)=2x3+3x2−12x−8f(x) = 2x^3 + 3x^2 - 12x - 8

We need to ensure that for all x∈[−2,2]x \in [-2, 2], the following holds:

−k≤f(x)≤k-k \leq f(x) \leq k

This implies:

∣f(x)∣≤kfor all x∈[−2,2]|f(x)| \leq k \quad \text{for all } x \in [-2, 2]

So, the smallest such kk is:

k=max⁡x∈[−2,2]∣f(x)∣k = \max_{x \in [-2, 2]} |f(x)|

Thus, our task reduces to finding the maximum absolute value of f(x)f(x) on the interval [−2,2][-2, 2].

Step 2: Find Critical Points

To find the extrema of f(x)f(x) on [−2,2][-2, 2], we compute the derivative:

f′(x)=6x2+6x−12f'(x) = 6x^2 + 6x - 12

Set f′(x)=0f'(x) = 0:

6x^2 + 6x - 12 = 0 \Rightarrow x^2 + embody a step-by-step solution to clearly show the reasoning and obtain the final result. --- ### Step 3: Evaluate the Function at Key Points We now evaluate $ f(x) $ at the critical points and endpoints of the interval $ [-2, 2] $. #### Critical Points: Solve $ f'(x) = 0 \Rightarrow x = -2 $ and $ x = 1 $ - $ f(-2) = 2(-2)^3 + 3(-2)^3(-2)^2 - 12(-2) - 8 = -16 + 12 + 24 - 8 = 12 $ - $ f(1) = 2(1)^3 + 3(1)^3(1)^2 - 12(1) - 8 = 2 + 3 - 12 - 8 = -15 $ - $ f(2) = 2(8) + 3(4) - 12(2) - 8 = 16 + 12 - 24 - 8 = -4 $ Also, check a few more points within the interval to confirm no other extrema: - $ f(-1) = -2 + 3 + 12 - 8 = 5 $ - $ f(0) = -8 $ - $ f(1.5) = 2(3.375) + 3(2.25) - 18 - 32 = 6.75 + 6.75 - 18 = -12.5 $ So the values are: - $ f(-2) = 12 $ - $ f(-1) = 5 $ - $ f(0) = -8 $ - $ f(1) = -15 $ - $ f(1.5) = -12.5 $ - $ f(2) = -4 $ --- ### Step 4: Determine Maximum Absolute Value Now compute the absolute values of these: - $ |f(-2)| = 12 $ - $ |f(-1)| = 5 $ - $ |f(0)| = 8 $ - $ |f(1)| = 15 $ - $ |f(1.5)| = 12.5 $ - $ |f(2)| = 4 $ The **maximum** of these is $ \boxed{15} $, achieved at $ x = 1 $, where $ f(1) = -15 $. --- ### Step 5: Conclusion Since $ |f(x)| \leq 15 $ for all $ x \in [-2, 2] $, and it's **not possible** for $ |f(x)| $ to be less than 15 (as shown by evaluation), the **smallest possible** value of $ k $ satisfying the inequality is:

\boxed{15}

Reasoning Process

[THOUGHT] I need to find the smallest positive k such that for all x∈[−2,2]x \in [-2, 2], the inequality −k≤2x3+3x2−12x−8≤k-k \leq 2x^3 + 3x^2 - 12x - 8 \leq k holds. This requires finding the maximum absolute value of the function f(x)=2x3+3x2−12x−8f(x) = 2x^3 + 3x^2 - 12x - 8 on the interval [−2,2][-2, 2].

To do this, I'll:

  1. Find critical points by computing f′(x)=6x2+6x−12f'(x) = 6x^2 + 6x - 12
  2. Evaluate f(x)f(x) at critical points and endpoints
  3. Determine the maximum absolute value of these values

Let me use Python to compute these values accurately. [/THOUGHT] [PYTHON] import numpy as np

def f(x): return 2x**3 + 3x**2 - 12*x - 8

Endpoints

endpoints = [-2, 2]

Critical points: solve f'(x) = 0 => 6x^2 + 6x - 12 = 0 => x^2 + x - 2 = 0 => x = -2, x = 1

critical_points = [-2, 1]

Combine all points to evaluate

points = list(set(endpoints + critical_points)) print("Points to evaluate:", points)

Evaluate f(x) at all points

values = [f(x) for x in points] print("f(x) values:", values)

Calculate absolute values

abs_values = [abs(v) for v in values] print("Absolute values:", abs_values)

Find the maximum absolute value

max_abs = max(abs_values) print("Maximum absolute value:", max_abs)

Also check a few intermediate points to be thorough

x_test = np.linspace(-2, 2, 1000) f_values = [f(x) for x in x_test] max_abs_full = max(abs(v) for v in f_values)

print("Maximum absolute value over continuous interval:", max_abs_full) </PYTHON>

Tool Output
[PYTHON OUTPUT] SyntaxError: invalid syntax (<string>, line 34) [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] I need to verify the calculation by evaluating the function f(x) = 2x^3 + 3x^2 - 12x - 8 at key points in [-2, 2], particularly at the critical points and endpoints, and find the maximum absolute value. Let me write a correct Python code for this calculation. [/THOUGHT] [PYTHON] import numpy as np

def f(x): return 2x**3 + 3x**2 - 12*x - 8

Critical points: solve f'(x) = 6x^2 + 6x - 12 = 0

This gives x^2 + x - 2 = 0, so (x+2)(x-1) = 0

So critical points are x = -2 and x = 1

critical_points = [-2, 1]

Endpoints

endpoints = [-2, 2]

Combine all points to evaluate

points = list(set(critical_points + endpoints))

Evaluate f(x) at all points

values = [f(x) for x in points] abs_values = [abs(v) for v in values]

print("Points:", points) print("f(x) values:", values) print("Absolute values:", abs_values) print("Maximum absolute value:", max(abs_values)))

Double-check by sampling more points

x_test = np.linspace(-2, 2, 1000) f_values = [f(x) for x in x_test] max_abs = max(abs(v) for v in f_values) print("Maximum absolute value (full sampling):", max_abs) [/PYTHON]

Tool Output
[PYTHON OUTPUT] SyntaxError: unmatched ')' (<string>, line 24) [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] I need to calculate the maximum absolute value of f(x) = 2x^3 + 3x^2 - 12x - 8 on the interval [-2, 2] to find the smallest k that satisfies the inequality. Let me write a correct Python implementation. [/THOUG con el fin de getting accurate results, let me rewrite the Python code more carefully.

import numpy as np

def f(x):
    return 2*x**3 + 3*x**2 - 12*x - 8

# Critical points: solve f'(x) = 6x^2 + 6x - 12 = 0
# Divide by 6: x^2 + x - 2 = 0
# Factor: (x+2)(x-1) = 0
# So critical points are x = -2 and x = 1
critical_points = [-2, 1]

# Endpoints
endpoints = [-2, 2]

# Combine all points to evaluate
points = list(set(critical_points + endpoints))

# Evaluate f(x) at all points
values = [f(x) for x in points]
abs_values = [abs(v) for v in values]

print("Points:", points)
print("f(x) values:", values)
print("Absolute values:", abs_values)
print("Maximum absolute value:", max(abs_values))

# Double-check by sampling more points
x_test = np.linspace(-2, 2, 1000)
f_values = [f(x) for x in x_test]
max_abs_full = max(abs(v) for v in f_values)
print("Maximum absolute value (full sampling):", max_abs_full)

Wait, I need to be more careful. Let me trace through this step by step:

f(x) = 2x^3 + 3x^2 - 12x - 8

Critical points: f'(x) = 6x^2 + 6x - 12 = 6(x^2 + x - 2) = 6(x+2)(x-1) = 0 So critical points are x = -2 and x = 1.

Endpoints are x = -2 and x = 2.

So the key points are x = -2, x = 1, and x = 2.

Let me evaluate f(x) at these points: f(-2) = 2(-8) + 3(4) - 12(-2) - 8 = -16 + 12 + 24 - 8 = 12 f(1) = 2(1) + 3(1) - 12(1) - 8 = 2 + 3 - 12 - 8 = -15 f(2) = 2(8) + 3(4) - 12(2) - 8 = 16 + 12 - 24 - 8 = -4

So the absolute values are: |f(-2)| = 12, |f(1)| = 15, |f(2)| = 4

The maximum absolute value is 15.

Let me verify this with Python: </think>

[THOUGHT] Let me write a correct Python implementation to verify the calculations for finding the maximum absolute value of f(x) = 2x^3 + 3x^2 - 12x - 8 on [-2, 2]. [/THOUGort the complexity, I realize the issue is in my previous approach. Let me rewrite the code correctly, paying attention to syntax and proper indentation.

I need to:

  1. Define the function f(x)
  2. Find critical points by solving f'(x) = 0
  3. Evaluate f(x) at critical points and endpoints
  4. Calculate the maximum absolute value

Let me write a correct Python implementation. </think>

[THOUGHT] Let me carefully write the correct Python code to calculate the maximum absolute value of the function f(x) = 2x^3 + 3x^2 - 12x - 8 on the interval [-2, 2]. [/THOUGHT] [PYTHON] import numpy as np

def f(x): return 2x**3 + 3x**2 - 12*x - 8

Critical points: solve f'(x) = 6x^2 + 6x - 12 = 0

Divide by 6: x^2 + x - 2 = 0

Factor: (x+2)(x-1) = 0

So critical points are x = -2 and x = 1

critical_points = [-2, 1]

Endpoints

endpoints = [-2, 2]

Combine all points to evaluate

points = list(set(critical_points + endpoints))

Evaluate f(x) at all points

values = [f(x) for x in points] abs_values = [abs(v) for v in values]

print("Points:", points) print("f(x) values:", values) print("Absolute values:", abs_values) print("Maximum absolute value:", max(abs_values))

Double-check by sampling more points

x_test = np.linspace(-2, 2, 1000) f_values = [f(x) for x in x_test] max_abs_full = max(abs(v) for v in f_values) print("Maximum absolute value (full sampling):", max_abs_full)

Tool Output
[PYTHON OUTPUT] Points: [1, 2, -2] f(x) values: [-15, -4, 12] Absolute values: [15, 4, 12] Maximum absolute value: 15 Maximum absolute value (full sampling): 14.999990983978975 [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] Based on the Python output, the maximum absolute value of f(x) = 2x^3 + 3x^2 - 12x - 8 on the interval [-2, 2] is 15, which occurs at x = 1 where f(1) = -15. Therefore, the smallest positive k such that the inequality holds for all x is k = 15. [/THOUGHT] \boxed{15}

Final Model Output
15

Comparative Analysis (P31) (Average across runs)

Avg Time Taken

EXAONE-4.0.1-32B (high)
2301.7
Solar-Open-100B
640.8
K-EXAONE-236B-A23B
189.6
Gemini-3-Pro-Preview
170.7
Llama-VARCO-8B-Instruct
98.0
Deepseek-V3.2
67.5
HCX-007(high)
54.8
Solar-Pro-2 (31B)(high)
35.0
Kanana-2-30B-Thinking
33.8
Kanana-2-30B-Thinking-2601
21.2
A.X-4.0 (72B)
15.2
GPT-5.2 (high)
14.8
GPT-5.1 (high)
13.3
Claude-Opus-4.5
12.6
K-EXAONE-236B-A23B
12.4
Grok-4.1-fast
10.0
axk1
4.4

Avg Token Usage

EXAONE-4.0.1-32B (high)
37496.0
K-EXAONE-236B-A23B
27042.0
Kanana-2-30B-Thinking
7460.0
Solar-Pro-2 (31B)(high)
7096.0
HCX-007(high)
5642.0
Kanana-2-30B-Thinking-2601
5493.5
Solar-Open-100B
5057.0
Llama-VARCO-8B-Instruct
4580.0
GPT-5.2 (high)
4360.0
Gemini-3-Pro-Preview
4189.0
K-EXAONE-236B-A23B
4115.0
Grok-4.1-fast
3931.0
Deepseek-V3.2
3561.0
Claude-Opus-4.5
3451.0
A.X-4.0 (72B)
3174.0
GPT-5.1 (high)
3116.0
axk1
2308.0