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Question #41391

Difficulty: 8/10 | 🟢 Free

The Question

Consider a right circular cylinder with base radius $r(x) = x^2 - 9x + 18$ and height $h(x) = e^{-x}$ . The volume $V(x)$ is defined only for those $x \in \mathbb{R}$ where the cylinder has strictly positive dimensions. Let $S$ be the set of all $x$ where $V(x)$ attains a local maximum. Find the sum of the elements in $S$ .

Options

A.
9
B.
13
C.
4
D.
18

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

Geometry Constraint Modeling|Logarithmic Differentiation|Logarithmic Differentiation Domain

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate geometric constraints with calculus optimization.

The primary trap lies in the domain of the volume function.

Students commonly apply logarithmic differentiation to find the critical points of the algebraic expression $V(x)$ , yielding local maxima at both $x=4$ and $x=9$ .

However, they often fail to verify these points against the physical domain of the cylinder.

At $x=4$ , the radius $r(x)$ evaluates to a negative value, making it geometrically impossible.

By ignoring the implicit constraint that physical dimensions must be strictly positive, students will incorrectly include the extraneous solution $x=4$ and arrive at the wrong sum of 13.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Geometry Constraint Modeling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free