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Question #41860

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the region in the $xy$ -plane defined by the system of inequalities $y \ge 0$ and $y \le -x^2 + 4x - 3$ . Let $x_1$ and $x_2$ ( $x_1 < x_2$ ) be the abscissae of the intersection points of the curves $y = 0$ and $y = -x^2 + 4x - 3$ .

A function $f:\mathbb{R} \to \mathbb{R}$ is defined as $f(x) = \begin{vmatrix} |x^2 - 4x + 3| & x^2 - 4x + 3 & 1 \\ x^2 & 2x & 1 \\ 1 & 0 & x \end{vmatrix}$

Find the value of $f'(x_1^+) + f'(x_2^-)$ .

Options

A.
-88
B.
-24
C.
-56
D.
-16

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

System of Inequalities|Differentiation of Determinants|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the differentiation of determinants containing piecewise functions, specifically focusing on the 'Modulus Sign Dependency' trap.

Students often mechanically differentiate $|g(x)|$ as $g'(x)$ or incorrectly apply the signum function without analyzing the specific directional limits ( $x_1^+$ and $x_2^-$ ).

By defining the points of interest via a system of inequalities, the problem forces the evaluation of the derivative from inside the region where $g(x) < 0$ .

Failing to flip the sign of the derivative of the modulus term leads to the trap answer -24, while incorrectly assuming the derivative is zero at the roots leads to -56.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	System of Inequalities
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free