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Question #40532

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \int_0^x \left( [t] - \frac{3}{2} \right) \left( [t] - \frac{7}{2} \right) (t-2)^2 (t-4)^4 \, dt$ , where $[ \cdot ]$ denotes the greatest integer function. Let $S$ be the sum of the squares of the $x$ -coordinates of all points of local extremum of $f(x)$ in the interval $(0, 6)$ . Then the value of $S$ is

Options

A.
0
B.
10
C.
20
D.
26

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

Even Multiplicity Roots|Critical Point vs Stationary Point|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the student's reliance on the standard calculus heuristic that even multiplicity roots of a derivative (like $(x-2)^2$ and $(x-4)^4$ ) do not produce local extrema because they do not change sign.

By introducing the Greatest Integer Function terms, a simultaneous sign change is forced exactly at these roots.

Students who blindly apply the even-power rule will incorrectly conclude there are no extrema (yielding 0).

To succeed, students must analyze the piecewise constant nature of the GIF and realize it provides the necessary sign change, distinguishing between continuous algebraic sign invariance and discrete step-function jumps.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Even Multiplicity Roots
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free