Before You Begin

Question #40681

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = x^2 - 4$ and $g(x) = |x-3| - 4$ for all $x \in \mathbb{R}$ . Define $H(x) = f(g(x))$ . Let $S$ be the set of all real numbers $x$ such that $H'(x) = 0$ , and $D$ be the set of all real numbers $x$ such that $H(x)$ is not differentiable. If $\alpha = \sum_{x \in S} x$ , $\beta = n(S)$ , and $\gamma = n(D)$ , where $n(A)$ denotes the number of elements in set $A$ , find the value of $\alpha^2 + \beta^2 + \gamma^2$ .

Options

A.
41
B.
49
C.
90
D.
91

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

Selection of AP Terms|Rolle's Theorem Anchors|Chain Rule Inner Function Differentiability

Why This Question Is Tricky

Common Mistakes

This question tests the limits of Rolle's Theorem on composite functions with absolute values.

The roots of $H(x) = 0$ are $\{-3, 1, 5, 9\}$ , forming an Arithmetic Progression.

A common trap is blindly applying Rolle's Theorem to assume $H'(x) = 0$ has roots in every interval between these zeros, leading to the false conclusion that $S = \{-1, 3, 7\}$ .

However, Rolle's Theorem requires differentiability on the open interval.

Here, $H(x)$ has a cusp at $x=3$ (exactly inside the interval $(1, 5)$ ), meaning the derivative never vanishes there.

Students who rely on memorized AP patterns for derivatives of polynomials will fail to verify the domain of differentiability, incorrectly inflating the size of set $S$ and missing the non-differentiable point $D$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Selection of AP Terms
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free