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Question #40718

Difficulty: 8/10 | 🟢 Free

The Question

Let a sequence of $n$ real numbers $A_1, A_2, \dots, A_n$ be inserted between $A_0 = 0$ and $A_{n+1} = 0$ . The terms of this sequence are generated by the function $f_n(x) = x(n+1-x)(x-4)^{1/3}$ such that $A_k = f_n(k)$ for $k=0, 1, \dots, n+1$ . Let $S$ be the set of all integers $n \in [1, 10]$ for which Rolle's Theorem is applicable to $f_n(x)$ on the interval $[0, n+1]$ . Let $T$ be the set of all integers $n \in [1, 10]$ for which $f_n(x)$ has exactly three critical points in the open interval $(0, n+1)$ . Find the ordered pair $\left( \sum_{n \in S} n, \sum_{n \in T} n \right)$ .

Options

A.
$(6, 49)$
B.
$(55, 0)$
C.
$(6, 0)$
D.
$(55, 49)$

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

Insertion of Means|Rolle's Theorem Anchors|Critical Point vs Stationary Point

Why This Question Is Tricky

Common Mistakes

The question tests the nuanced distinction between stationary points (where $f'(x)=0$ ) and critical points (where $f'(x)=0$ or is undefined), embedded within the application of Rolle's Theorem.

Students often mechanically apply Rolle's Theorem by only checking $f(a)=f(b)$ and continuity, missing the strict requirement of differentiability on the open interval, which fails here due to the fractional power creating a non-differentiable point at $x=4$ .

Furthermore, the 'Lethal Collision' occurs when identifying critical points: students typically equate the derivative to zero and solve the resulting quadratic, finding two roots.

They frequently forget that the point where the derivative is undefined ( $x=4$ ) is also a valid critical point, leading them to incorrectly conclude that the function only has two critical points instead of three for $n \ge 4$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Insertion of Means
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free