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Question #41367

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = x^7 + 3x^5 + 8x^3 + 2$ . Let $g(x)$ be the inverse function of $f(x)$ . If $H(x) = g(x^3 - 3x^2 + 4)$ , find the sum of the squares of all critical points of $H(x)$ .

Options

A.
9
B.
13
C.
4
D.
17

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

Critical Point vs Stationary Point|Inverse Function Derivative|Chain Rule Complexity

Why This Question Is Tricky

Common Mistakes

This question targets the conceptual difference between critical points and stationary points in the context of composite inverse functions.

Students often equate critical points solely with $H'(x) = 0$ , successfully finding $x=0$ and $x=2$ .

However, they fall into the trap of ignoring points where $H'(x)$ is undefined.

By using the inverse function theorem $g'(y) = 1/f'(g(y))$ , the derivative $g'(y)$ is undefined when $f'(x) = 0$ .

This creates non-differentiable critical points (vertical tangents) for $H(x)$ that must be included in the final sum.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Critical Point vs Stationary Point
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free