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Question #41352

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \int_{0}^{x} (4t + 5\text{sgn}(t-1)) \, dt$ , where $\text{sgn}(\cdot)$ denotes the signum function. Let $m_1$ and $m_2$ be the left-hand and right-hand derivatives of $f(x)$ at $x=1$ respectively. Let $L_1$ and $L_2$ be the lines passing through the point $(1, f(1))$ with slopes $-\frac{1}{m_1}$ and $-\frac{1}{m_2}$ respectively. If $x_1$ and $x_2$ are the x-intercepts of $L_1$ and $L_2$ , find the value of $x_1 x_2$ .

Options

A.
-104
B.
121
C.
16
D.
-26

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

Tangent/Normal Slope Interplay|Newton-Leibniz Formula with variable limits|Differentiability vs Continuity

Why This Question Is Tricky

Common Mistakes

This question tests the interplay between the Newton-Leibniz formula and the differentiability of functions defined by integrals.

The core trap lies in blindly applying the Leibniz rule to find the derivative at $x=1$ .

Since $\text{sgn}(0) = 0$ , a student might incorrectly calculate $f'(1) = 4(1) + 5(0) = 4$ , leading to a single erroneous normal line with an x-intercept of $-11$ , yielding an incorrect product of $121$ .

The problem forces the student to recognize that the integrand is discontinuous at $t=1$ , meaning $f(x)$ has a sharp corner (kink) at $x=1$ .

Consequently, one must evaluate the left-hand and right-hand derivatives separately by analyzing the piecewise nature of the signum function before and after the limit point.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Tangent/Normal Slope Interplay
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free