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Question #70145

Difficulty: 8/10 | 🟢 Free

The Question

Let $y(x)$ be the positive solution to the differential equation $\frac{dy}{dx} + \frac{1}{2} y = \frac{1}{2} (x^5 - 15x^4 + 75x^3 - 160x^2 + 150x - 20) y^3$ for $x > 0$ , with the initial condition $y(0) = \frac{1}{2\sqrt{5}}$ . Let $S$ be the set of all points $x \in (0, \infty)$ where $y(x)$ attains a local extremum. Let $k = \sum_{x \in S} x$ . Consider the polynomial $f(x) = x^3 - k x^2 + 11x - 6$ . Which of the following statements is true?

Options

A.
$f(x)$ satisfies the conditions of Rolle's Theorem on $[1, 3]$
B.
$f(x)$ satisfies the conditions of Rolle's Theorem on $[1, 4]$
C.
$f(x)$ satisfies the conditions of Rolle's Theorem on $[2, 4]$
D.
$f(x)$ does not satisfy the conditions of Rolle's Theorem on any interval $[a, b]$ for $a, b \in \{1, 2, 3, 4\}$

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

Bernoulli Equation Transformation|Rolle's Theorem Anchors|First Derivative Test Trap

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the integration of Bernoulli differential equations with advanced calculus concepts, specifically targeting a common trap in the First Derivative Test.

After transforming the non-linear DE into a linear one using $v = y^{-2}$ , students find the derivative $v'(x) = 5(x-1)^2(x-2)(x-4)$ .

The trap lies in assuming that all roots of $v'(x) = 0$ correspond to local extrema.

Students who fail to check the sign change of $v'(x)$ will incorrectly include the stationary point $x=1$ in the set $S$ , leading to an incorrect sum $k=7$ .

This cascading error causes them to evaluate the wrong polynomial $f(x)$ for Rolle's Theorem, effectively filtering out candidates who rely on rote memorization over rigorous analytical verification.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Bernoulli Equation Transformation
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free