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Question #40346

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = 6x - x^2$ . A geometric constraint function is defined as $D(x) = |f(x) - x|$ , representing the vertical distance to the identity line. Let $H(x) = D(D(x))$ . Let $S$ be the set of all real numbers $x$ where $H(x)$ is not differentiable. Find the sum of the squares of the elements in $S$ .

Options

A.
75
B.
50
C.
65
D.
85

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

Identity Function Behavior|Geometry Constraint Modeling|Concavity Analysis|Chain Rule Inner Function Differentiability

Why This Question Is Tricky

Common Mistakes

This question tests the differentiability of composite functions involving absolute values, specifically modeling a geometric constraint against the identity line.

The primary trap is the 'inner function cusp' trap.

Students typically identify the non-differentiability of $D(x)$ at $x=0$ and $x=5$ , but fail to apply the chain rule condition $H'(x) = D'(D(x))D'(x)$ .

This requires finding points where the inner function's value hits the outer function's non-differentiable points (i.e., $D(x) = 5$ ).

Ignoring this leads to missing 4 critical points, testing both formal limit definition and geometric intuition.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Identity Function Behavior
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free