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Question #41368

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x)$ be a differentiable function such that $f'(x) = (x-2)^2 (x-3)(x-4)$ . Let $S$ be the set of all points of local extrema of $f(x)$ . If $A$ is the area of the region bounded by the curve $y = f'(x)$ , the x-axis, and the vertical lines $x = \min(S)$ and $x = \max(S)$ , then find the value of $60A$ .

Options

A.
16
B.
23
C.
30
D.
7

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

Critical Point vs Stationary Point|Arbitrary Constants vs Parameters|Bounded vs Unbounded Regions|Even Multiplicity Roots

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between stationary points and local extrema, specifically targeting the 'even multiplicity' trap.

Students often mistakenly assume that all roots of $f'(x)=0$ are local extrema.

Here, $x=2$ is a root of even multiplicity, meaning the derivative does not change sign, making it an inflection point rather than a local extremum.

If a student incorrectly includes $x=2$ in the set $S$ , they will evaluate the integral over the wrong interval $[2, 4]$ , leading to the trap answer 30.

The problem further integrates the application of definite integrals for bounded regions, requiring careful sign analysis of the integrand to correctly compute the area.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Critical Point vs Stationary Point
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free