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Question #60500

Difficulty: 8/10 | 🟢 Free

The Question

Let the sequences $\{\alpha_n\}$ and $\{\beta_n\}$ be defined by $\alpha_1 = 2$ , $\beta_1 = -2$ , and for $n \ge 1$ : $\alpha_{n+1} = \frac{\alpha_n^2 + 1}{2\alpha_n}, \quad \beta_{n+1} = \frac{\beta_n^2 + 1}{2\beta_n}$ Let $P_n(x) = (x - \alpha_n)(x - \beta_n)$ be a sequence of polynomials. Define the function $f: \mathbb{R} \to \mathbb{R}$ by $f(x) = \lim_{n \to \infty} \frac{x^{2n} - 1}{x^{2n} + 1} P_n(x)$ Let $g(x) = f(x) + \lambda x^2$ , where $\lambda \in \mathbb{R}$ . Which of the following represents the complete set of values of $\lambda$ for which $g(x)$ has exactly three distinct points of local extrema?

Options

A.
(-1, 1)
B.
$(-\infty, -1) \cup (1, \infty)$
C.
[-1, 1]
D.
(-1, 0) \cup (0, 1)

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

Polynomial Continuity/Differentiability Link, Quadratic Roots in Sequences, First Derivative Test Trap, Unbounded Region Trap

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

The question tests the student's ability to link sequence limits to piecewise polynomial functions and identify extrema using the First Derivative Test while avoiding the Unbounded Region Trap.

Students often erroneously assume uniform convergence, ignoring the boundary behavior at $|x|=1$ , leading them to differentiate $g(x) = (\lambda+1)x^2 - 1$ globally.

This yields only one critical point ( $x=0$ ).

To find all three extrema, the student must correctly evaluate the pointwise limit to obtain the non-differentiable function $f(x) = |x^2 - 1|$ , and then apply the First Derivative Test by checking the sign changes of $g'(x)$ across the boundary points $x = \pm 1$ , rather than merely solving $g'(x) = 0$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Polynomial Continuity/Differentiability Link
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free