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Question #41227

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: [2, \infty) \to [-4, \infty)$ be defined by $f(x) = x^2 - 4x$ . Let $g: \mathbb{R} \to [-4, \infty)$ be defined by $g(x) = x^2 - 6|x| + 5$ . Let $h(x) = f^{-1}(g(x))$ . If $S$ is the set of all real numbers where the derivative $h'(x)$ does not exist, then the value of $\sum_{x \in S} x^2 + \int_{-4}^{4} h(x) dx$ is:

Options

A.
44
B.
50
C.
8
D.
26

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Composition Inverse Identity Traps|Inverse Function Derivative|Geometrical Interpretation of Modulus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply the identity $\sqrt{x^2} = |x|$ when composing an inverse function with a restricted domain.

A common JEE Advanced trap is assuming $f^{-1}(f(y)) = y$ universally, leading students to incorrectly write $\sqrt{(|x|-3)^2} = |x|-3$ .

This ignores the outer modulus, which is mathematically invalid because the range of $f^{-1}$ is restricted to $[2, \infty)$ , meaning $h(x)$ must be $\ge 2$ .

Missing this outer modulus changes the non-differentiable points (missing $\pm 3$ ) and drastically alters the area under the curve, leading directly to the trap option 8.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Composition Inverse Identity Traps
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free