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Question #66244

Difficulty: 8/10 | 🟢 Free

The Question

Consider the following two statements:

Statement I: The function $f(x) = \begin{cases} \frac{x^4 - 16}{x - 2}, & x \le 2 \\ 3x^2 + 12x - 4, & x > 2 \end{cases}$ is differentiable at $x = 2$ .

Statement II: For the given function $f(x)$ , $\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x)$ and $\lim_{x \to 2^-} f'(x) = \lim_{x \to 2^+} f'(x)$ .

Choose the correct option:

Options

A.
Statement I is True, Statement II is True; Statement II is a correct explanation for Statement I
B.
Statement I is True, Statement II is True; Statement II is NOT a correct explanation for Statement I
C.
Statement I is True, Statement II is False
D.
Statement I is False, Statement II is True

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

Differentiability vs Continuity|Rational Function Asymptotes|Denominator Restriction

Why This Question Is Tricky

Examiner Levers

Logical Assertion

Tests the causal relationship between two true statements.

Difficulty Boost : +4

Common Mistakes

This question exploits the 'Denominator Restriction' trap in piecewise functions.

Students commonly rely on the shortcut of equating left-hand and right-hand limits of both the function and its derivative to check for differentiability.

While $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$ and $\lim_{x \to c^-} f'(x) = \lim_{x \to c^+} f'(x)$ hold true in this scenario, students fail to verify if the function is actually defined at the junction point.

Here, the equality condition $x \le 2$ forces the evaluation of $f(2)$ into the rational expression, yielding a $0/0$ indeterminate form.

Since the function is undefined at the point, it is neither continuous nor differentiable, rendering Statement I false despite the perfectly matching limits.

Quick Facts About This Question

Question Information

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