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Question #40810

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = |x^3 - 12x|$ . Let $S$ be the set of all $x$ -coordinates of the points on the curve $y = f(x)$ at which the tangent line to the curve has a slope $m$ , where $m \\in \\{-36, -24, -12, 0, 12, 24, 36\\}$ . Find the number of elements in $S$ .

Options

A.
6
B.
7
C.
8
D.
9

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

Condition of Tangency|Composition Continuity|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the 'Modulus Breakdown' trap.

Students often incorrectly assume the composition $f(x) = |g(x)|$ is globally differentiable and blindly solve for tangency by taking $\\pm g'(x) = m$ .

They fail to verify if the resulting $x$ -coordinates fall within the correct domain branch of the modulus, and crucially, they fail to reject the roots of $g(x)$ where the modulus function forms a sharp corner and is non-differentiable.

A careless student will include the non-differentiable points $0$ and $\\pm 2\\sqrt{3}$ , leading to the trap answer of 9 elements.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Condition of Tangency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free