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Question #41862

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The Question

Let $S$ be the solution set of the system of inequalities $|2x - 3| \le 1$ and $x^2 - 4 \le 0$ . Let $f: S \to \mathbb{R}$ be defined by $f(x) = \begin{cases} (x-1)^2 \sin\left(\frac{1}{x-1}\right) + x^2, & x \in (1, 2] \\ 1, & x = 1 \end{cases}$ Let $A = f'(1^+)$ be the right-hand derivative of $f$ at $x=1$ . Let $B$ be the value of $f'(c)$ for the point $c \in (1, 2)$ guaranteed by Lagrange's Mean Value Theorem for $f$ on $S$ . If $A$ exists, find the value of $A + B$ . (If $A$ does not exist, select the corresponding option).

Options

A.
$\sin(1) + 5$
B.
$\sin(1) + 3$
C.
\text{Does not exist}
D.
$2\sin(1) - \cos(1) + 6$

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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

System of Inequalities|Mean Value Theorem Applicability|The L'Hôpital's Rule Misuse

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of Lagrange's Mean Value Theorem (LMVT) and the definition of differentiability, specifically targeting the misuse of L'Hôpital's rule.

The domain $S$ is restricted to $[1, 2]$ via a system of inequalities.

Students must find the right-hand derivative $A = f'(1^+)$ .

A common trap is applying L'Hôpital's rule to evaluate the limit $\lim_{x \to 1^+} \frac{f(x)-f(1)}{x-1}$ , which requires evaluating $\lim_{x \to 1^+} f'(x)$ .

Because the derivative contains an oscillating $\cos(1/(x-1))$ term, this limit does not exist, leading students to falsely conclude that $f$ is non-differentiable at $x=1$ or that $A$ does not exist.

However, L'Hôpital's rule is only a sufficient condition; the limit must be evaluated using the first principle, which correctly yields $A=2$ .

Furthermore, the question reinforces that LMVT only requires differentiability on the open interval $(1, 2)$ , not at the boundaries.

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