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Question #40736

Difficulty: 8/10 | 🟢 Free

The Question

Consider the curve defined parametrically by $x(t) = t^3 + t$ and $y(t) = |t|$ for $t \in [-1, 2]$ . Since $x(t)$ is strictly increasing, the curve can be represented by an explicit function $y = f(x)$ on the interval $[-2, 10]$ . Let $S$ be the set of all $x \in (-2, 10)$ where the tangent to the curve is parallel to the secant line joining its endpoints. Let $P$ be the logical proposition "Lagrange's Mean Value Theorem is applicable to $f(x)$ on $[-2, 10]$ ". Which of the following options is correct?

Options

A.
$P$ is False, and $S = \left\{ \frac{14}{3}\sqrt{\frac{11}{3}} \right\}$
B.
$P$ is True, and $S = \left\{ \frac{14}{3}\sqrt{\frac{11}{3}} \right\}$
C.
$P$ is False, and $S = \emptyset$
D.
$P$ is True, and $S = \emptyset$

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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

Parametric Form Manipulation|LMVT Tangent Parallelism|Non-Differentiable Extrema

Why This Question Is Tricky

Common Mistakes

This question tests the strict prerequisites of Lagrange's Mean Value Theorem (LMVT) alongside parametric differentiation.

Students often blindly apply the formula $f'(c) = \frac{f(b)-f(a)}{b-a}$ without verifying differentiability across the entire open interval.

By embedding $y(t) = |t|$ , a non-differentiable sharp turn is created at $x=0$ , rendering LMVT inapplicable.

The trap is assuming that the failure of LMVT implies the non-existence of a parallel tangent.

The student must manually solve the parametric derivative to find the valid point, proving that while the theorem's guarantee fails, the geometric condition can still be met.

Quick Facts About This Question

Question Information

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