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Question #41305

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = (x^2 - 6x + 8)^{4/5}$ be a function defined on the interval $[0, 6]$ . Let $m$ be the number of points in the open interval $(0, 6)$ where $f(x)$ is not differentiable. Let $n$ be the number of points $c \in (0, 6)$ such that $f'(c) = 0$ . Let $k = 1$ if Rolle's Theorem is applicable to $f(x)$ on $[0, 6]$ , and $k = 0$ otherwise. Find the value of $2^m + 3^n + 5^k$ .

Options

A.
8
B.
9
C.
12
D.
6

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

The Zero-Factor Cancellation Trap|Rolle's Theorem Anchors|Closed vs Open Intervals

Why This Question Is Tricky

Common Mistakes

This question tests the 'Zero-Factor Cancellation Trap' within the hypothesis verification of Rolle's Theorem.

Students frequently differentiate functions with fractional powers and focus exclusively on the numerator to find a point $c$ where $f'(c) = 0$ , ignoring the domain restrictions imposed by the denominator.

By doing so, they overlook the points where the denominator vanishes ( $x=2$ and $x=4$ ), which are points of non-differentiability.

Because these points lie strictly within the open interval $(0, 6)$ , the fundamental differentiability condition for Rolle's Theorem is violated, rendering it inapplicable ( $k=0$ ).

The trap successfully catches students who find a valid $c=3$ and falsely conclude that the theorem's applicability is verified, leading them to incorrect values for $m$ and $k$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	The Zero-Factor Cancellation Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free