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Question #41906

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: [246, 248] \rightarrow \mathbb{R}$ be defined by $f(x) = |x^2 - 496x + 61503|$ . Let $S$ be the set of all $c \in (246, 248)$ such that $f$ is differentiable at $c$ and $f'(c) = \frac{f(248) - f(246)}{2}$ . The number of elements in $S$ is:

Options

A.
0
B.
1
C.
2
D.
3

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Modulus Function Breakdown|Mean Value Theorem Applicability|The L'Hôpital's Rule Misuse

Why This Question Is Tricky

Common Mistakes

This question tests the strict applicability of the Mean Value Theorem (MVT) and the piecewise differentiation of modulus functions.

A common fatal error is to blindly apply MVT or differentiate the inner function without respecting the modulus, which yields $c = 247.5$ .

Since $247.5 \in (246, 248)$ , students fall into the trap of assuming this $c$ is valid and answer 1.

However, MVT guarantees a solution only if the function is differentiable on the entire open interval.

Because $f(x)$ has a 'kink' at $x = 247$ , MVT is inapplicable, and piecewise analysis reveals no valid $c$ exists.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Modulus Function Breakdown
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free