Before You Begin

Question #40502

Difficulty: 8/10 | 🟢 Free

The Question

Let $P(x) = x^3 + ax^2 + bx + c$ be a polynomial with real coefficients. A function $f(x)$ is defined as $f(x) = P(x) \lfloor x \rfloor$ for $x \in [0, 2]$ , where $\lfloor \cdot \rfloor$ denotes the greatest integer function. If $f(x)$ satisfies all the conditions of Rolle's Theorem on the interval $[0, 2]$ , then the value of $a^2 + b^2 + c^2$ is:

Options

A.
45
B.
29
C.
38
D.
54

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

Condition for Common Roots|Rolle's Theorem Root Existence|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

The question tests the rigorous application of Rolle's Theorem in the presence of a step function discontinuity.

The primary trap is assuming that $P(x)$ only needs to be zero at the endpoints.

Students often overlook that for $f(x) = P(x)\lfloor x \rfloor$ to be differentiable at the integer jump $x=1$ , the polynomial must not only evaluate to zero ( $P(1)=0$ ) to ensure continuity, but its derivative must also be zero ( $P'(1)=0$ ) to match the zero gradient from the left-hand interval $[0, 1)$ .

This forces a common root between the polynomial and its derivative, making $x=1$ a repeated root.

Failing to partition the interval and check left/right differentiability leads to an underdetermined system for the coefficients.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Condition for Common Roots
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free