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Question #40730

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: [-59, 13] \to \mathbb{R}$ be defined as $f(x) = \sqrt[3]{(x-5)^2}$ . Let $S$ be the set of all points $c \in (-59, 13)$ such that the tangent to the curve $y = f(x)$ at $x = c$ is parallel to the chord joining the points $(-59, f(-59))$ and $(13, f(13))$ . Then the number of elements in $S$ is:

Options

A.
0
B.
1
C.
2
D.
3

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

Slope Sensitivity|Mean Value Theorem Applicability|Differentiability vs Continuity

Why This Question Is Tricky

Common Mistakes

This question tests the strict applicability conditions of the Mean Value Theorem (MVT) and the definition of the open interval for the point c.

Students often blindly assume MVT guarantees a point c where f'(c) equals the secant slope, forgetting to check if f(x) is differentiable on the entire open interval (here, it fails at x=5).

Furthermore, if they attempt to solve f'(c) = m algebraically, they find exactly one root (c = -59).

The lethal trap is that this root is the boundary endpoint, not inside the open interval (-59, 13).

Careless students will count this root and incorrectly answer 1, falling for a double-layered conceptual and algebraic trap.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Slope Sensitivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free