Before You Begin

Question #41229

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be a real-valued continuous function defined on $[-1, 1]$ satisfying the implicit equation $y^3 + e^y = \sin^{-1}(\sin|x|) + \cos^{-1}(\cos x) + 1$ Let $S$ be the set of all points $c \in (-1, 1)$ such that the tangent to the curve $y = f(x)$ at $x = c$ is parallel to the chord joining the points $(-1, f(-1))$ and $(1, f(1))$ . Find the number of elements in $S$ .

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

Derivative of Implicit Inverse Forms|LMVT Tangent Parallelism|The Modulus Trap|Symmetry and Periodicity

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of the Lagrange Mean Value Theorem (LMVT) and Rolle's Theorem.

A common trap is to observe $f(-1) = f(1)$ and blindly conclude via Rolle's Theorem that at least one point $c$ exists where $f'(c) = 0$ .

However, the presence of the modulus inside the inverse trigonometric functions creates a cusp at $x=0$ .

Students fail by ignoring the non-differentiability at the origin, which renders the LMVT application invalid.

The implicit form further obscures the derivative, ensuring only those who systematically verify existence criteria will correctly deduce that no such point exists.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Derivative of Implicit Inverse Forms
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free