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Question #40955

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: D \to \mathbb{R}$ be a real-valued function defined by $f(x) = \ln\left( \sqrt{x^2 - 25} - x \right) - \frac{x}{6}$ , where $D$ is the maximal possible domain of $f$ . If $m$ is the slope of the normal to the curve $y = f(x)$ at $x = 13$ , then the value of $m$ is:

Options

A.
4
B.
$-\frac{1}{4}$
C.
-4
D.
\text{Does not exist}

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

Chain Rule Complexity|Slope Sensitivity|Compound Function Boundaries

Why This Question Is Tricky

Common Mistakes

This question tests the student's vigilance regarding the domain of composite functions before applying differential calculus.

The specific trap leverages the mechanical application of the chain rule.

If a student blindly differentiates the function, the algebraic expression simplifies neatly to $-1/\sqrt{x^2 - 25} - 1/6$ , which evaluates to a clean, finite value ( $-1/4$ ) at $x = 13$ , leading to a normal slope of $4$ .

However, the logarithmic and square root constraints restrict the function's domain strictly to $(-\infty, -5]$ .

Because $x = 13$ is outside this domain, the function does not exist there, making the calculation of a tangent or normal slope mathematically invalid.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Chain Rule Complexity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free