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Question #41392

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all points $(a, b)$ lying on the curve $a^2 + b^2 - 6a - 8b = 0$ such that the function $f(x) = \frac{x^3}{3} - ax^2 + (a^2 - b^2 + 17b - 72)x + 2024$ is strictly increasing on $\mathbb{R}$ . Find the maximum value of $a+b$ for $(a, b) \in S$ .

Options

A.
14
B.
$7 + 5\sqrt{2}$
C.
12
D.
15

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Geometry Constraint Modeling|Increasing/Decreasing Strictness|Inconsistency Detection

Why This Question Is Tricky

Common Mistakes

This problem deploys the 'Constraint-Satisfiability' trap, specifically targeting the Domain-Existence fallacy in optimization.

Students typically use parametric coordinates or calculus to maximize $a+b$ subject to the circular geometric constraint, yielding the unconstrained maximum $7+5\sqrt{2}$ (Option B).

However, they fail to intersect this geometric locus with the analytic constraint derived from the strict monotonicity of $f(x)$ .

The condition $f'(x) \ge 0$ forces $b \in [8, 9]$ .

The unconstrained maximum occurs at $b = 4 + \frac{5}{\sqrt{2}} \approx 7.535$ , which lies outside this valid interval, creating a lethal collision between the geometric optimum and the analytic domain.

By recognizing this inconsistency, the student must optimize $a+b$ strictly on the boundary arc of the circle where $b \ge 8$ , leading to the true constrained maximum.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Geometry Constraint Modeling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free