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Question #41206

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x) = \begin{pmatrix} 0 & x-1 & 0 \\ 0 & 0 & x \\ (x-1)^2 & 0 & 0 \end{pmatrix}$ . A curve $C: y = f(x)$ passes through the point $(2, 1)$ . The slope of the tangent to $C$ at any point $(x, f(x))$ is given by $\frac{1}{3}\text{tr}(A(x)^3)$ . Let $S_1$ be the set of all $x$ -coordinates where $f(x)$ attains a local minimum, and $S_2$ be the set of all $x$ -coordinates where $f(x)$ attains a local maximum. Find the value of $20 \left( \sum_{x \in S_1} f(x) - \sum_{x \in S_2} f(x) \right)$ . (If a set is empty, its sum is taken as $0$ ).

Options

A.
-1
B.
1
C.
-12
D.
11

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Idempotent and Involutory Properties|Tangent/Normal Slope Interplay|Second Derivative Test Limitations

Why This Question Is Tricky

Common Mistakes

This question targets a classic vulnerability in calculus applications: the over-reliance on the Second Derivative Test.

By engineering the derivative $f'(x) = x(x-1)^3$ , the second derivative vanishes at the local minimum $x=1$ ( $f''(1)=0$ ).

Students who blindly apply the second derivative test will incorrectly classify $x=1$ as a point of inflection, leading them to assume $S_1$ is empty and select the trap option -12.

The problem forces the use of the First Derivative Test or higher-order analysis, while simultaneously testing matrix multiplication and trace properties as a rigorous entry barrier.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Idempotent and Involutory Properties
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free